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This study examines the equation of state and properties of neutron-rich nuclear matter and neutron stars using central heavy-ion reactions. It explores the symmetry energy at sub-saturation and supra-saturation densities, as well as implications for core-crust transitions and gravitational waves from elliptically deformed pulsars.

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  1. Bao-An Li & collaborators: Wei-Zhou Jiang, Plamen Krastev, Richard Nobra, Will Newton, De-Hua Wen and Aaron Worley,Texas A&M University-Commerce Lie-Wen Chen, Hongru Ma, Shanghai Jiao-Tung University Che-Ming Ko and Jun Xu, Texas A&M University, College Station Andrew Steiner, Michigan State University Zhigang Xiao and Ming Zhang, Tsinghua University, China Gao-Chan Yong and Xunchao Zhang, Institute of Modern Physics, China Constraining the EOS of neutron-rich nuclear matterand properties of neutron stars with central heavy-ion reactions • Outline: • Indication on the symmetry energy at sub-saturation densities from the NSCL/MSU isospin diffusion data • Astrophysical implications:(1) Core-crust transition density of neutron stars • (2) Gravitational waves from elliptically deformed pulsars • Indication on the symmetry energy at supra-saturation densities from the FOPI/GSI π-/π+ data • Summary

  2. What is the Equation of State in the extended isospin space? (EOS of neutron-rich matter) symmetry energy Isospin asymmetry δ ρn : neutron density ρp : proton density Nucleon density ρ=ρn+ρp 12 12 12 Energy per nucleon in symmetric matter 18 18 3 Energy per nucleon in asymmetric matter Symmetric matter ρn=ρp density ??? 0 ρ=ρn+ρp δ ??? 1 Isospin asymmetry

  3. The Esym (ρ) from model predictions using popular interactions Examples: EOS of pure neutron matter Alex Brown, PRL85, 5296 (2000). 23 RMF models ρ APR - Density

  4. The multifaceted influence of the isospin dependence of strong interactionand symmetry energy in nuclear physics and astrophysicsJ.M. Lattimer and M. Prakash, Science Vol. 304 (2004) 536-542.A.W. Steiner, M. Prakash, J.M. Lattimer and P.J. Ellis, Phys. Rep. 411, 325 (2005). (Effective Field Theory) (QCD) isodiffusion n/p Isospin physics π-/π+ isotransport in isocorrelation Terrestrial Labs isofractionation t/3He K+/K0 isoscaling

  5. Symmetry energy and single nucleon potential used in theIBUU04 transport model The x parameter is introduced to mimic various predictions by different microscopic nuclear many-body theories using different effective interactions stiff ρ soft Default: Gogny force Density ρ/ρ0 Single nucleon potential within the HF approach using a modified Gogny force: The momentum dependence of the nucleon potential is a result of the non-locality of nuclear effective interactions and the Pauli exclusion principle C.B. Das, S. Das Gupta, C. Gale and B.A. Li, PRC 67, 034611 (2003). B.A. Li, C.B. Das, S. Das Gupta and C. Gale, PRC 69, 034614; NPA 735, 563 (2004).

  6. Momentum dependence of the isoscalar potential Compare with variational many-body theory

  7. Momentum and density dependence of the symmetry (isovector) potential Lane potential extracted from n/p-nucleus scatterings and (p,n) charge exchange reactions provides only a constraint at ρ0: P.E. Hodgson, The Nucleon Optical Model, World Scientific, 1994 G.W. Hoffmann and W.R. Coker, PRL, 29, 227 (1972). G.R. Satchler, Isospin Dependence of Optical Model Potentials, in Isospin in Nuclear Physics, D.H. Wilkinson (ed.), (North-Holland, Amsterdam,1969)

  8. Constraints from both isospin diffusion and n-skin in 208Pb Isospin diffusion data: M.B. Tsang et al., PRL. 92, 062701 (2004); T.X. Liu et al., PRC 76, 034603 (2007) Transport model calculations B.A. Li and L.W. Chen, PRC72, 064611 (05) ρ ρρ J.R. Stone PREX Hartree-Fock calculations A. Steiner and B.A. Li, PRC72, 041601 (05) Neutron-skin from nuclear scattering: V.E. Starodubsky and N.M. Hintz, PRC 49, 2118 (1994); B.C. Clark, L.J. Kerr and S. Hama, PRC 67, 054605 (2003)

  9. Neutron Star Crust 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 density and pressure at the transition to calculate the fractional moment of inertia of the curst core crust Kazuhiro Oyamatsu, Kei Iida Phys. Rev. C75 (2007) 015801 Can one extract transition density from heavy-ion collisions? Chuck Horowitz at WCI3, Texas, 2005 Yes, the symmetry energy constrained by the isospin diffusion experiments at the NSCL is in the same density range of the inner crust

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

  11. What we found about the core-crust transition density It is NOT accurate 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: Thermodynamical method

  12. Constraint on the core-crust transition density Transition pressure pasta Need to reduce the error bars with more precise data and calculations! Kazuhiro Oyamatsu, Kei Iida Phys. Rev. C75 (2007) 015801 Jun Xu, Lie-Wen Chen, Bao-An Li and HongRu Ma, nucl-th (2008)

  13. Partially constrained EOS for astrophysical studies Bao-An Li, Lie-Wen Chen and Che Ming Ko Physics Reports, 464, 113 (2008) (flow, Danielewicz, Lacey and Lynch, Science 298, 1592 (2002))

  14. Gravitational waves from elliptically deformed pulsars Solving linearized Einstein’s field equation of General Relativity, the leading contribution to the GW is the mass quadrupole moment Frequency of the pulsar Distance to the observer Breaking stain of crust Mass quadrupole moment EOS B. Abbott et al., PRL 94, 181103 (05) B.J. Owen, PRL 95, 211101 (05)

  15. The ellipticity of pulsars EOS Plamen Krastev, Bao-An Li and Aaron Worley, Phys. Lett. B668, 1 (2008).

  16. Constraining the strength of gravitational wavesPlamen Krastev, Bao-An Li and Aaron Worley, Phys. Lett. B668, 1 (2008). Compare with the latest upper limits from LIGO+GEO observations Phys. Rev. D 76, 042001 (2007) It is probably the most uncertain factor B.J. Owen, PRL 95, 211101 (05)

  17. Pion ratio probe of symmetry energyat supra-normal densities GC Coefficients2

  18. Isospin asymmetry reached in heavy-ion reactions 48 48 E/A=800 MeV, b=0, t=10 fm/c 124 124 197 197

  19. t=10 fm/c t=10 fm/c Correlation between the N/Z and the π-/ π+ (distance from the center of the reaction system)

  20. Formation of dense, asymmetric nuclear matter Symmetry energy Stiff Central density Soft density π-/ π+ probe of dense matter Soft Esym Stiff Esym n/p ratio at supra-normal densities

  21. π-/π+ ratio as a probe of symmetry energy at supra-normal densities W. Reisdorf et al. for the FOPI/GSI collaboration , NPA781 (2007) 459 IQMD: Isospin-Dependent Quantum Molecular Dynamics C. Hartnack, Rajeev K. Puri, J. Aichelin, J. Konopka, S.A. Bass, H. Stoecker, W. Greiner Eur. Phys. J. A1 (1998) 151-169 Need a symmetry energy softer than the above to make the pion production region more neutron-rich! low(high)density region is more neutron-rich with stiff (soft)symmetry energy

  22. Soft symmetry energy FRIB? APR Stiff symmetry energy

  23. ? MSU-TPC?

  24. For pure nucleonic matter The softest symmetry energy that the TOV is still stable is x=0.93 giving m_max=0.1 and R=40 km

  25. Can the symmetry energy becomes negative at high densities? Yes, due to the isospin-dependence of the nuclear tensor force The short-range repulsion in n-p pair is stronger than that in pp and nn pairs At high densities, the energy of pure neutron matter can be lower than symmetric matter leading to negative symmetry energy Example: proton fraction with 10 interactions leading to negative symmetry energy

  26. In hyperonic matter Asymmetric nuclear matter

  27. Summary • Based on the NSCL/MSU data, the symmetry energy at sub-saturation densities is constrained to • The FOPI/GSI pion data indicates a symmetry energy at supra-saturation densities • softer than the APR prediction

  28. Comparing with calculations using 23 most popular RMF models L.W. Chen, C.M. Ko and B.A. Li, PRC 76, 054316 (2007)

  29. Isospin-dependence of nucleon-nucleon cross sectionsin neutron-rich matter in neutron-rich matter at zero temperature The effective mass scaling model: is the reduced effective mass of the colliding nucleon pair NN according to Dirac-Brueckner-Hatree-Fock calculations F. Sammarruca and P. Krastev, nucl-th/0506081 Applications in symmetric nuclear matter: 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). Application in neutron-rich matter: nn and pp xsections are splitted due to the neutron-proton effective mass slitting Bao-An Li and Lie-Wen Chen, nucl-th/0508024, Phys. Rev. C72, 064611 (2005).

  30. How to determine experimentally the isospin-dependence of in-medium NN xsections Traditional measures of stopping power using global observables, such as, quadrupole moment, LMT, ERAT, etc, are sensitive to the values of the in-medium NN xsections, but they are ambiguous for extracting the isospin-dependence of the NN xsections. An example: quadrupole moment QZZ σnp/σpp=1.5-3 σnp/σpp=1 Insensitive to the isospin-dependence of the NN xsection because the total no. of NN collisions are about the same:

  31. Neutron-proton effective k-mass splitting in neutron-rich matter With the modified Gogny effective interaction B.A. Li, C.B. Das, S. Das Gupta and C. Gale, PRC 69, 034614 (2004); NPA 735, 563 (2004).

  32. Nucleon effective k-masses during heavy-ion reactionsB.APhys. Rev. . Li and L.W. Chen, C72, 064611 (2005). Instant of maximum compression Effective mass distributions

  33. Astrophysical impacts of the partially constrained symmetry energy Nuclear constraints on the moment of inertia of neutron stars Aaron Worley, Plamen Krastev and Bao-An Li, The Astrophysical Journal (2008) in press . Constraining properties of rapidly rotating neutron stars using data from heavy-ion collisions Plamen Krastev, Bao-An Li and Aaron Worley, The Astrophysical Journal, 676, 1170 (2008) Constraining time variation of the gravitational constant G with terrestrial nuclear laboratory data Plamen Krastev and Bao-An Li, Phys. Rev. C76, 055804 (2007). Constraining the radii of neutron stars with terrestrial nuclear laboratory data Bao-An Li and Andrew Steiner, Phys. Lett. B642, 436 (2006). Nuclear limit on gravitational waves from elliptically deformed pulsars Plamen Krastev, Bao-An Li and Aaron Worley, Phys. Lett. B668, 1 (2008). Locating the core-crust transition point in neutron stars with nuclear laboratory data, Jun Xu, Lie-Wen Chen, Bao-An Li and HongRu Ma (2008)

  34. Possible sources of Gravitational Waves: Compact binary inspiral: “chirps” Examples Supernovae / GRBs: “bursts” Orbital decay of the Hulse-Taylor binary neutron star system (Nobel prize in 1993) is the best, but indirect evidence so far. Elliptically deformed pulsars:“periodic” Non-radial oscillations of neutron stars

  35. Estimate of gravitational waves from spinning-down of pulsars Assumption: spinning-down is completely due to the GW radiation “Standard fiducial value” • Solid black lines: LIGO and GEO science requirement, for T=1 year • Circles: upper limits on gravitational waves from known EM pulsars, obtained from measured spindown • Only known, isolated targets shown here GEO LIGO The LIGO Scientific Collaboration, Phys. Rev. D 76, 042001 (2007)

  36. Testing the standard fudicial value of the moment of inertia Aaron Worley, Plamen Krastev and Bao-An Li, The Astrophysical Journal (2008) in press .

  37. Constraining the mass-radius relation of fast pulsarsPlamen Krastev, Bao-An Li and Aaron Worley, The Astrophysical Journal, 676, 1170 (2008) Solving the Einstein equation in general relativity using the RNS code written by Nikolaos Stergioulas and John L. Friedman, Astrophysics J. 444, 306 (1995)

  38. (completely due to general relativity)

  39. Neutron star matter equation of state and gravitational wave emission Authors: Omar Benhar Mod.Phys.Lett. A20 (2005) 2335-2350

  40. EOS of neutron-rich matter enters here: The first w-mode The frequency is inversely proportional to the compactness of the star compactness of the star Mon.Not.Roy.Astron.Soc. 299 (1998) 1059-1068

  41. The f-mode Fit calculations using 12 different EOSs

  42. B.A. Brown PRL85, 5296 (2000) S. Typel and B.A. Brown, PRC 64, 027302 (2001) Rn-Rp (fm) for 208Pb Neutron-skin in 208Pb and dEsym/dρ B.A. Brown, S. Typel, C. Horowitz, J. Piekarewicz, R.J. Furnstahl, J.R. Stone, A. Dieperink et al. C.J. Horowitz and J. Piekarewicz, PRL 86, 5647 (2001) 208Pb Pressure forces neutrons out against the surface tension from the symmetric core near ρ0 Neutron-skin

  43. Parity Radius Experiment (PREX)Jefferson National Laboratory spokespersons: R. Michaels, P. Souder, G. Urciuoli Parity violation probes neutron density of heavy nucleus and measures density dependence of symmetry energy. PREX references: http://cecelia.physics.indiana.edu/prex Chuck Horowitz 208Pb

  44. Constraining the radii of non-rotating neutron stars Bao-An Li and Andrew W. Steiner, Phys. Lett. B642, 436 (2006) . ● Nuclear limits APR: K0=269 MeV. The same incompressibility for symmetric nuclear matter of K0=211 MeV for x=0, -1, and -2

  45. Astronomers discover the fastest-spinning neutron-star spining at 716 Science 311, 1901 (2006).

  46. The proton fraction x at ß-equilibrium in proto-neutron stars is determined by The critical proton fraction for direct URCA process to happen is Xp=0.14 for npeμ matter obtained from energy-momentum conservation on the proton Fermi surface Slow cooling: modified URCA: Consequence: long surface thermal emission up to a few million years Faster cooling by 4 to 5 orders of magnitude: direct URCA PSR J0205+6449 in 3C58 was suggested as a candidate Bao-An Li, Phys. Rev. Lett. 88, 192701 (2002).

  47. Extract the Esym(ρ) at subnormal densities from isospin diffusion/transport A quantitative measure of isospin transport: for complete isospin mixing X is any isospin-sensitive observable, F. Rami et al. (FOPI/GSI), PRL, 84 (2000) 1120. The degree of isospin transport/diffusion depends on both the symmetry potential and the in-medium neutron-proton scattering cross section. Isospin transport/diffusion: For near-equilibrium systems, the mean-field contributes: L. Shi and P. Danielewicz, PRC68, 064604 (2003) During heavy-ion reactions, the collisional contribution to DI is expected to be proportional to σnp

  48. Isospin transport/diffusion experiments at the NSCL/MSU M.B. Tsang et al., Phys. Rev. Lett. 92, 062701 (2004); T.X. Liu et al., PRC 76, 034603 (2007)    X7=7Li/7Be

  49. Symmetry energy from isoscaling analyses X=0 D.V. Shetty, S.J. Yennello and G.A. Souliotis Phys. Rev. C75 (2007) 034602; Phys. Rev. C76 (2007) 024606

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