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									 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. lwchen@sjtu.edu.cn)


  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)


 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24-30日,赤峰,内蒙古
                               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, 113-281 (2008)
L.W. Chen, B.J. Cai, C.M. Ko, B.A. Li, C. Shen, and J. Xu, PRC80, 014322 (2009)
L.W. Chen, C.M. Ko, B.A. Li and J. Xu, arXiv:1004.4672, 2010
                                 EOS of asymmetric nuclear matter and
                                 the symmetry energy
                              Isospin in Nuclear Physics
   On Earth!!!       Transport Theory                       General Relativity In Heaven!!!
                                             EOS for
Isospin Effects in HIC’s …                  Asymmetric               Neutron Stars …
                                           Nuclear Matter

                         Many-Body Theory                                  Reactions &
  Most uncertain
                                                                           Structures
  property of an                            Nuclear Force                  of Neutron-
   asymmetric
                                                                           Rich Nuclei
  nuclear matter         Many-Body Theory                                 (CSR/Lanzho
                             Structures of Neutron-rich Nuclei, …         u, FRIB, GSI,
                                                                            RIKEN……)

                                       Density Dependence of
                                   the Nuclear Symmetry Energy


                              Isospin Nuclear Physics
  What is the isospin dependence of the in-medium nuclear effective interactions???
                             EOS of Nuclear Matter
The energy of per nucleon in a nuclear matter with density  , temperature T, and
                            n - p
isospin asymmetry  (               ) can be expressed as
                      E / A     (  , T ,  ) (Nuclear Matter EOS)
The pressure P of the nuclear matter can be expressed as
                                                
                         P(  , T ,  )   2     
                                                T , N constant
The incompessibilty K of the nuclear matter can be expressed as
                                          P 
                         K (,T , )  9     
                                           T , N constant
Empirical values about the nuclear matter EOS:
Saturation density (P0 ( 0 )  0) of symmetric nuclear matter at T=0 MeV:  0  0.16 fm 3
The energy of per nucleon of symmetric nuclear matter at 0 and T=0 MeV:
 0  16 MeV/nucleon
Incompessibilty of symmetric nuclear matter at T=0 MeV: K 0  200        400 MeV
                       The Nuclear Symmetry Energy

                          Liquid-drop model



                                                  (Isospin)
                                             Symmetry energy term




                Symmetry energy including surface diffusion effects (y s=Sv/Ss)

W. D. Myers, W.J. Swiatecki, P. Danielewicz, P. Van Isacker, A. E. L. Dieperink,……
                                           The Nuclear Matter Symmetry Energy
                             EOS of Isospin Asymmetric Nuclear Matter (Parabolic law)
                       E( ,  )  E( ,0)  Esym ( ) 2  O( 4 ),   (n   p ) / 
                                   The Nuclear Symmetry Energy
                                      1 2 E(, )
Symmetric Nuclear Matter Esym (  )                                                              Symmetry energy term
(relatively well-determined)          2     2                                                      (poorly known)
                                                                             2
                                      K sym     0 
                                L    0
     Esym (  )  Esym (  0 )                     , (  0 )
                                3  0
                                        18   0 
     Esym (  0 )  30 MeV (LD mass formula: Myers & Swiatecki, NPA81; Pomorski & Dudek, PRC67 )
                Esym (  )
     L  3 0                           (Many-Body Theory: L : 50                200 M eV; Exp: ???)
                               0

                        2 Esym (  )
     K sym  9    2
                                                 (Many-Body Theory: K sym : 700             466 MeV; Exp: ???)
                             2
                   0
                                          0

     The isospin part of the isobaric incompressiblity K of asymmetric nuclear matter
     K  K sym  6 L  J 0 / K 0 L (GMR : 320  180 MeV (Sharma et al., PRC38, 2562 (88));
                                                    566  1350       34  159 MeV(Shlomo&Youngblood,PRC47,529(93);
                                                    550  100 MeV(T. Li et al, PRL99,162503(2007)))
                             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
 ……
                       Esym: Many-Body Approaches


Chen/Ko/Li, PRC72, 064309(2005)             Chen/Ko/Li, PRC76, 054316(2007)
  Z.H. Li et al., PRC74, 047304(2006)   Dieperink et al., PRC68, 064307(2003)

                    BHF
                        Symmetry energy around saturation density

                      Promising Probes of the Esym(ρ)
                          (an incomplete list !)
At sub-saturation densities (亚饱和密度行为)
  Sizes of n-skins of unstable nuclei from total reaction cross sections
  Proton-nucleus elastic scattering in inverse kinematics
  Parity violating electron scattering studies of the n-skin in 208Pb
  n/p ratio of FAST, pre-equilibrium nucleons
  Isospin fractionation and isoscaling in nuclear multifragmentation
  Isospin diffusion/transport
  Neutron-proton differential flow
  Neutron-proton correlation functions at low relative momenta
  t/3He ratio
  Hard photon production

Towards high densities reachable at CSR/Lanzhou, FAIR/GSI, RIKEN,
GANIL and, FRIB/MSU (高密度行为)
  π -/π + ratio, K+/K0 ratio?
  Neutron-proton differential transverse flow
  n/p ratio at mid-rapidity
  Nucleon elliptical flow at high transverse momenta
  n/p ratio of squeeze-out emission
                                     Esym: Isospin Diffusion in HIC’s
     Symmetry energy, isospin diffusion, in-medium cross section
                                                Chen/Ko/Li, PRL94,032701 (2005)
Isospin dependent BUU transport model
                                                Li/ Chen, PRC72, 064611(2005)
   Chen/Ko/Li, PRC72,064309 (2005)




  Fit the symmetry energy with                 Isospin Diffusion Data 
  Esym (  )  31.6(  / 0 ) MeV             Esym(ρ0)=31.6 MeV
  (From 0    0 ), we obtain:                  L=88±25 MeV
    1.05 for x  1 and   0.69 for x  0
                             Esym: Isoscaling in HIC’s
Constraining Symmetry Energy by Isocaling: TAMU Data
 Shetty/Yennello/ Souliotis, PRC75,034602(2007); PRC76, 024606 (2007)




                                    Isoscaling Data 
                                    Esym(ρ0)=31.6 MeV
                                    L=65 MeV

                                   Consistent with isospin diffusion data!
           Esym: Isospin diffusion and double n/p ratio in HIC’s

ImQMD: n/p ratios and two isospin diffusion measurements
Tsang/Zhang/Danielewicz/Famiano/Li/Lynch/Steiner, PRL 102, 122701 (2009)




                                         ImQMD: Isospin Diffusion and
                                         double n/p ratio 
                                         Esym(ρ0)=28 - 34 MeV
                                         L=38 - 103 MeV
               Esym: Nuclear Mass in Thomas-Fermi Model
                Myers/Swiatecki, NPA 601, 141 (1996)
Thomas-Fermi Model analysis of 1654 ground state mass of nuclei with N,Z≥8




Thomas-Fermi Model + Nuclear Mass  Esym(ρ0)=32 .65 MeV L=49.9 MeV
                             Esym: Pygmy Dipole Resonances




Pygmy Dipole Resonances of 130,132Sn  Esym(ρ0)=32 ± 1.8 MeV L=43.125 ± 15 MeV




Pygmy Dipole Resonances of 68Ni and 132Sn 
Esym(ρ0)=32.3 ± 1.3 MeV, L=64.8 ± 15.7 MeV
                                   Esym: IAS+LDM

                 Danielewicz/Lee, NPA 818, 36 (2009)
Esym from Isobaric Analog States + Liquid Drop model with surface
                        symmetry energy




          IAS+Liquid Drop Model with Surface Esym 
          Esym(ρ0)=32.5 ± 1 MeV L=94.5 ± 16.5 MeV
                        Esym: Droplet Model Analysis on Neutron Skin




Droplet Model + N-skin  Esym(ρ0)=31.6 MeV, L=66.5 ± 36.5 MeV
              Esym: Droplet Model Analysis on Neutron Skin




Droplet Model + N-skin  Esym(ρ0)=28 - 35 MeV, L=55 ± 25 MeV
                                 Esym around normal density
9 constraints on Esym (ρ0) and L from nuclear reactions and structures




                                                           Esym(ρ0)=28 - 35 MeV
                                                           L=28 - 111 MeV


                                                 Still within large uncertain region !!
                 The Nuclear Neutron Skin

                                                       Bodmer,
                                               Nucl. Phys. 17, 388 (1960)

                                              Sprung/Vallieres/Campi/Ko,
                                                  NPA253, 1 (1975)

                                                   Shlomo/Friedman,
                                                   PRL39, 1180 (1977)

                                                             ……




                                           2 1/2       2 1/2
For heavier stable nuclei: N>Z            r
                                          n         r p

                                       2 1/2         2 1/2
       Neutron Skin Thickness: rnp  r
                                      n         - r  p
                          The Esym vs. Nuclear Neutron Skin

                            Chen/Ko/Li, PRC72,064309 (2005)




            Neutron-Skin Thickness:           Good linear Correlation: S-L
            S    rn2     rp2   (fm)

S ( 208 Pb) varies from 0.04 fm to 0.24 fm
depending on the Skyrme interaction !
                          The Esym vs. Nuclear Neutron Skin
                           Chen/Ko/Li, PRC72,064309 (2005)




            For heavier nuclei: Still good linear correlation between S-L
Neutron-Skin Thickness: S                       S    ( pn  p p )   L
Pressure difference between n and p: pn  p p
Slope of the Symmetry Energy: L                 B.A. Brown, PRL85,5296 (2000)
                  The Skyrme HF Energy Density Functional
                    Standard Skyrme Interaction:
                                                   There are more than
                                                   120 sets of Skyrme-
                                                    like Interactions in
                                                        the literature


                                                   Agrawal/Shlomo/Kim Au
                                                    PRC72, 014310 (2005)

                            _________                 Yoshida/Sagawa
 Chen/Ko/Li/Xu                                      PRC73, 044320 (2006)
arXiv:1004.4672

9 Skyrme parameters:
9 macroscopic nuclear properties:
                  The Skyrme HF Energy Density Functional

Chen/Cai/Ko/Li/Shen/Xu, PRC80, 014322 (2009): Modified Skyrme-Like (MSL) Model




Chen/Ko/Li/Xu, arXiv:1004.4672
The Skyrme HF with MSL0

        Chen/Ko/Li/Xu, arXiv:1004.4672
                     Correlations between Nuetron-Skin thickness
                     and macroscopic Nuclear Properties




                  Important Terms


For heavy nuclei 208Pb and 120Sn:
Δrnp is strongly correlated with L, moderately with Esym(ρ0), a little bit with m*s,0
For medium-heavy nucleus 48Ca:
Δrnp correlation with Esym is much weaker; It further depends on GV and W0
Constraining Esym with Neutron Skin Data




                Neutron skin constraints on
                L and Esym(ρ0) are insensitive to
                the variations of other
                macroscopic quantities.
                       Constraining Esym with Neutron Skin Data
                                       and Heavy-Ion Reactions




N-Skin + HIC                                  (~independent of Esym(ρ0))
 Core-Crust transition density in      A quite stringent constraint on
 Neutron stars:                        Δrnp of 208Pb:
                               Esym: Global nucleon optical potential


    Xu/Li/Chen, arXiv:1006.4321v1, 2010




Global nucleon optical potential  Esym(ρ0)=31.3 ± 4.5 MeV, L=52.7 ± 22.5 MeV

                                  Consistent with Sn neutron skin data!
             Symmetry energy and Nuclear Effective Interaction
Chen/Ko/Li, PRC72,064309 (2005)   Chen/Ko/Li, PRC76, 054316(2007)




  L=58 ± 18 MeV: only 32/118        L=58 ± 18 MeV: only 8/23
                  IV. Summary and Outlook

 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 208Pb
Combining the constraints on Esym 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,,…… )
谢 谢!
                      EOS of Symmetric Nuclear Matter
          (1) EOS of symmetric matter around the saturation density ρ0
                             d 2E      Giant Monopole Resonance
 Incompressibility: K 0 =9 ( 2 ) 0
                          2

                             d
                          0




                                        Frequency fGMR  K0
K0=231±5 MeV
PRL82, 691 (1999)
Recent results:
K0=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 1ρ0< ρ < 3ρ0 from K+ production in HIC’s
                                                 J. Aichelin and C.M. Ko,
                                                    PRL55, (1985) 2661
                                                         C. Fuchs,
                                            Prog. Part. Nucl. Phys. 56, (2006) 1
                                                    C. Fuchs et al,
                                                 PRL86, (2001) 1974
                                              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.”
                                             See also: C. Hartnack, H. Oeschler,
                                                       and J. Aichelin,
                                                   PRL96, 012302 (2006)
                       EOS of Symmetric Nuclear Matter
(3) Present constraints on the EOS of symmetric nuclear matter for 2ρ0< ρ < 5ρ0 using
                     flow data from BEVALAC, SIS/GSI and AGS
P. Danielewicz, R. Lacey and W.G. Lynch, Science 298, 1592 (2002)
                                                     The highest pressure recorded under
                                                     laboratory controlled conditions in
                                                     nucleus-nucleus collisions




                                                                                      px

                                                                                           y




     Use constrained mean fields to predict the       High density nuclear matter
     EOS for symmetric matter                         2 to 5ρ0
       •   Width of pressure domain reflects
                                                                            E 
                                                      Pressure P( )   2 
                                                                               s
           uncertainties in comparison and of
           assumed momentum dependence.                                        
                           Transport model for HIC’s
        Isospin-dependent BUU (IBUU) model
Phase-space distributions f ( r , p, t ) satify the Boltzmann equation
       f ( r , p, t )
                         p   r f   r   p f  I c ( f ,  NN )
            t
   Solve the Boltzmann equation using test particle method
   Isospin-dependent initialization
   Isospin- (momentum-) dependent mean field potential
                            1
                    V  V0  (1   z )VC  Vsym            EOS
                            2
   Isospin-dependent N-N cross sections
   a. Experimental free space N-N cross section σexp
   b. In-medium N-N cross section from the Dirac-Brueckner
     approach based on Bonn A potential σin-medium
   c. Mean-field consistent cross section due to m*
   Isospin-dependent Pauli Blocking
                         Transport model: IBUU04
         Isospin- and momentum-dependent potential (MDI)

Das/Das Gupta/Gale/Li,
PRC67,034611 (2003)


                                                 MDI Interaction
                                                 (   Gogny)
                                                  0  0.16 fm 3
                                                 E (  0 ) / A  16 MeV
                                                 Esym (  0 )  31.6 MeV
                                                 K 0  211 MeV
                                                 m * / m  0.68
                                                 Chen/Ko/Li,
                                                 PRL94,032701 (2005)
                                                 Li/Chen,
                                                 PRC72, 064611 (2005)
                     Esym: Isospin Diffusion in HIC’s
                 Isospin Diffusion/Transport




______________________________________
How to measure                         PRL84, 1120 (2000)
Isospin Diffusion?                                A+A,B+B,A+B
                                                  X: isospin tracer
                               Esym: Isoscaling in HIC’s
                        Isoscaling in HIC’s
                  Isoscaling observed in many reactions
M.B. Tsang et al. PRL86, 5023 (2001)


                                                      Y2 / Y 
                                                            1
                                                          ( N n  Z  p ) / T
                                                      e
             High density behaviors of Esym
            Heavy-Ion Collisions at Higher Energies




                                  n/p ratio of the high density region


Isospin fractionation!           Li/Yong/Zuo, PRC 71, 014608 (2005)
                          High density behaviors of Esym: kaon ratio
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, 202301 (2006)     Exp.: Lopez et al. FOPI, PRC75, 011901(R) (2007)
                                                             96
                                                             44   Ru+ 96 Ru and
                                                                      44
                                                             96
                                                             40   Zr+ 96 Zr@1.528 AGeV
                                                                      40




Subthreshold K0/K+ yield may be a sensitive         K0/K+ yield is not so sensitive to the symmetry
probe of the symmetry energy at high densities      energy! Lower energy and more neutron-rich
                                                    system???
                       High density behaviors of Esym: pion ratio
                 A Quite Soft Esym at supra-saturation densities ???
IBUU04, Xiao/Li/Chen/Yong/Zhang, PRL102,062502(2009) Zhang et al.,PRC80,034616(2009)




                                                                Pion Medium Effects?
                                                                     Xu/Ko/Oh
                                                                PRC81, 024910(2010)

                                                                 Threshold effects?
                                                                       ……




 IDQMD, Feng/Jin, PLB683, 140(2010)
                      High density behaviors of Esym: n/p v2
              A Stiff Esym at supra-saturation densities ???
W. Trauntmann et al., arXiv:1001.3867
            Esym at very low densities: Clustering effects
Horowitz and Schwenk,
Nucl. Phys. A 776 (2006) 55




 S. Kowalski, et al.,
PRC 75 (2007) 014601.
                         Esym at very low densities: Clustering effects
J. B. Natowitz et al.,
   arXiv:1001.1102
      PRL, 2010

								
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