1. The Nuclear Symmetry Energy and Neutron Star Crusts Lie-Wen Chen ( 陈列文 ) (Department of Physics, Shanghai Jiao Tong University) Compact stars in the QCD phase diagram II May 20 ‐ 24, 2009, Beijing Collaborators ： Wei-Zhou Jiang (South-East U.) Che Ming Ko and Jun Xu (TAMU) Bao-An Li (TAMU-Commerce) Gao-Chan Yong (IMP,CAS) Hong-Ru Ma (SJTU) Zhi-Gang Xiao and Ming Zhang (Tsinghua U.)

2. Outline EOS of asymmetric nuclear matter and the nuclear symmetry energy Constraining the density dependence of the nuclear symmetry energy in heavy-ion collisions The nuclear symmetry energy and neutron star crusts Summary and outlook Main References: L.W. Chen, C.M. Ko, B.A. Li, and G.C. Yong, Front. Phys. China 2(3), 327 (2007) [arXiv:0704.2340] B.A. Li, L.W. Chen, and C.M. Ko, Phys. Rep. 464, 113-281 (2008) [arXiv:0804.3580] J. Xu, L.W. Chen, B.A. Li, and H.R. Ma, Phys. Rev. C 79, 035802 (2009) [arXiv:0807.4477] J. Xu, L.W. Chen, B.A. Li, and H.R. Ma, Astrophys. J. 697, 1549-1568 (2009) [arXiv:0901.2309]

3. I. EOS of Asymmetric Nuclear Matter and the Nuclear Symmetry Energy Neutron Stars … Structures of Radioactive Nuclei, SHE … Isospin Effects in HIC’s … Many-Body Theory Transport Theory General Relativity Nuclear Force EOS for Asymmetric Nuclear Matter Density Dependence of the Nuclear Symmetry Energy HIC’s induced by neutron-rich nuclei (CSR/Lanzho u,FRIB,GSI, RIKEN……) Most uncertain property of an asymmetric nuclear matter What is the isospin dependence of the in-medium nuclear effective interactions??? Isospin Physics in medium energy nuclear physics

4. Radioactive beam facilities are being built around the world IMP CIAE Providing new opportunities for both nuclear physics and astrophysics World status of Rare Isotope Accelerators

5. Many-Body Approaches to Nuclear Matter EOS 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 ……

6. Equation of State of symmetric nuclear matter is relatively well determined (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±20 MeV G. Colo et al. U. Garg et al. __

7. (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 Equation of State of symmetric nuclear matter is relatively well determined See also: C. Hartnack, H. Oeschler, and J. Aichelin, PRL96, 012302 (2006)

8. (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 Equation of State of symmetric nuclear matter is relatively well determined

9. Liquid-drop model 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

10. EOS of Asymmetric Nuclear Matter (Parabolic law) The Nuclear Symmetry Energy The Nuclear Matter Symmetry Energy Symmetry energy term Symmetric Nuclear Matter

11. Chen/Ko/Li, PRC72, 064309(2005) Chen/Ko/Li, PRC76, 054316(2007) The Nuclear matter symmetry energy Z.H. Li et al., PRC74, 047304(2006)Dieperink et al., PRC68, 064307(2003) BHF

12. II. Constraining the density dependence of the nuclear symmetry energy in heavy-ion collisions Promising Probes of the E sym (ρ) in Nuclear Reactions (an incomplete list !)

13. 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

14. Isospin- and momentum-dependent potential (MDI) Chen/Ko/Li, PRL94,032701 (2005) Li/Chen, PRC72, 064611 (2005) Das/Das Gupta/Gale/Li, PRC67,034611 (2003) Transport model: IBUU04

15. in neutron-rich matter is the reduced mass of the colliding pair NN in medium J.W. Negele and K. Yazaki, PRL 47, 71 (1981) V.R. Pandharipande and S.C. Pieper, PRC 45, 791 (1992) M. Kohno et al., PRC 57, 3495 (1998) D. Persram and C. Gale, PRC65, 064611 (2002). 1.In-medium cross sections are reduced 2.nn and pp cross sections are splitted due to the neutron-proton effective mass slitting in neutron-rich matter Li/Chen, PRC72 (2005)064611 Medium effects: effective mass on the incoming current in initial state and level density of the final state Neglecting medium effects on the transition matrix In-medium Nucleon-nucleon cross sections: Effective mass scaling model

16. (1) Isospin diffusion (2) Isospin scaling Chen/Ko/Li, PRL94(05); PRC72(05); Li/Chen, PRC72(05) Shetty et al. PRC75(07);PRC76(07) Zhang et al. PLB664(08) (3) Double n/p ratio Symmetry Energy: Sub-saturation density behaviors IBUU04 ImQMD

17. IBUU04 : S~31.6(  /  o )    ImQMD: Double n/p ratios and two isospin diffusion measurements Consistent constraints from the  2 analysis of three observables S=12.5(  /  o ) 2/3 + 17.6 (  /  o )  i  i   i   i  ii Tsang/Zhang/Danielewicz/Famiano/Li/Lynch/Steiner, PRL 102, 122701 (2009) Symmetry Energy: Sub-saturation density behaviors

18. (IBUU04) (ImQMD) X=-1 Chen/Ko/Li, PRL 94, 032701 (2005)Tsang et al., PRL 102, 122701 (2009) Symmetry Energy: Sub-saturation density behaviors

19. 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 (ZX Li, QF Li et al., M. Di Toro et al., …) Theory: Famiano et al., PRL97, 052701 (2006)Exp.: Lopez et al. FOPI, PRC75, 011901(R) (2007) K 0 /K + yield is not so sensitive to the symmetry energy! Lower energy and more neutron-rich system??? High density behaviors: Kaon Probe

20. IBUU04, Xiao/Li/Chen/Yong/Zhang, PRL102, 062502(2009) A Quite Soft Esym at supra-saturation densities !!! M. Zhang et al., arXiv:0904.0447 High density behaviors: Pion Probe

21. Lattimer/Prakash, Science 304, 536 (2004) Neutron star has solid crust over liquid core. Rotational glitches: small changes in period from sudden unpinning of superfluid vortices. Evidence for solid crust. 1.4% of Vela moment of inertia glitches. Needs to know the transition density to calculate the fractional moment of inertia of the crust Link et al., PRL83,3362 (99) III. The Nuclear Symmetry Energy and Neutron Star Crusts core-crust transition

22. 2. Thermodynamic approach Or, similarly one can use 3. the RPA If one uses the parabolic approximation (PA) Then the stability condition is: >0 Onset of instability in the uniform n+p+e matter 1. Dynamical approach k  0 (neglecting Coul.) Stability condition:

23. Core-Crust Transition Density: Parabolic Law fails! (1)It is NOT enough to know the symmetry energy, one almost has to know the exact EOS of n-rich matter Why? Because it is the determinant of the curvature matrix that determines the stability condition Example: Not so surprise: Zhang/Chen, CPL 18 (2000) 142 Steiner, Phys.Rev. C74 (2006) 045808 Higher-order term effects on direct URCA Xu/Chen/Li/Ma, PRC79, 035802 (2009)

24. (2) Locating the inner edge of neutron star crust Kazuhiro OyamatsuKazuhiro Oyamatsu, Kei IidaKei Iida Phys. Rev. C75 (2007) 015801 pasta Xu/Chen/Li/Ma, PRC79, 035802 (2009) Xu/Chen/Li/Ma, ApJ 697, 1547 (2009), arXiv:0901.2309 Parabolic Approximation has been assumed !!! Significantly less than their fiducial values: ρ t =0.07-0.08 fm -3 and P t =0.65 MeV/fm 3

25. (3) Constraints on M-R relation of NS (Isospin Diff) (Empirical estimate Link et al., PRL83,3362(99)) Lattimer Prakash

26. (4) Properties of neutron star crusts Xu/Chen/Li/Ma, ApJ 697, 1549 (2009), arXiv:0901.2309 Larger L leads to thicker neutron-skin, but thinner neutron star crust !!!

27. (5) Inner Crust EOS Dependence Xu/Chen/Li/Ma, ApJ 697, 1549 (2009), arXiv:0901.2309 The mass is insensitive to the inner crust EOS The radius is sensitive to the inner crust EOS for a softer symmetry energy The inner crust EOS has tiny effects on the dI/I when dI/I is small

28. The softest symmetry energy that the TOV is still stable is x=0.93 giving M_max=0.11 solar mass and R=>28 km For pure nucleonic matter??? New Physics??? Soft symmetry energy at HD ? K 0 =211 MeV is used, higher incompressibility for symmetric matter will lead to higher masses systematically ? (6) HD Esym and properties of neutron stars

29. The isospin diffusion data, Isoscaling and Isotope dependence of GMR seem to give a stringent constraint for the sub-saturation density behavior of the symmetry energy (L=86±25 MeV and K asy =-500±50 MeV) Probing the high density behavior of the symmetry energy remains a big challenge and the pion ratio data from FOPI favor a quite soft Esym at high densities. Significant constraints on inner edge and crust properties of neutron stars have been already obtained from present knowledge on symmetry energy at sub-saturation density region. Crosscheck is definitely needed !!! IV. Summary and Outlook

30. Thanks ！

31. Institute of Nuclear, Particle, Astronomy and Cosmology-INPAC, Shanghai Jiao Tong University http://physics.sjtu.edu.cn/iwdd09/index.html

32. International workshop on nuclear dynamics in heavy-ion reactions and the symmetry energy (IWND2009) August 22–25, Shanghai, China Organizers Shanghai Institute of Applied Physics, CAS Shanghai Jiao Tong University Beijing Normal University Institute of Modern Physics, CAS