Constraints on the skewness coefficient of symmetric nuclear matter within the nonlinear relativistic mean field model


Within the nonlinear relativistic mean field (NL-RMF) model, we show that both the pressure of symmetric nuclear matter at supra-saturation densities and the maximum mass of neutron stars are sensitive to the skewness coefficient, \(J_0\), of symmetric nuclear matter. Using experimental constraints on the pressure of symmetric nuclear matter at supra-saturation densities from flow data in heavy-ion collisions and the astrophysical observation of a large mass neutron star PSR J0348+0432, with the former favoring a smaller \(J_0\) while the latter favors a larger \(J_0\), we extract a constraint of \(-\,494\,\mathrm {MeV}\le J_0\le -\,10 \,\mathrm{MeV}\) based on the NL-RMF model. This constraint is compared with the results obtained in other analyses.

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  1. 1.

    J.P. Blaizot, Nuclear compressibilities. Phys. Rep. 64, 171–248 (1980).

    MathSciNet  Article  Google Scholar 

  2. 2.

    B.A. Li, C.M. Ko, W. Bauer, Isospin physics in heavy-Ion collisions at intermediate energies. Int. J. Mod. Phys. E 7, 147 (1998).

    Article  Google Scholar 

  3. 3.

    P. Danielewicz, R. Lacey, W.G. Lynch, Determination of the equation of state of dense matter. Science 298, 1592–1596 (2002).

    Article  Google Scholar 

  4. 4.

    V. Baran, M. Colonna, V. Greco et al., Reaction dynamics with exotic nuclei. Phys. Rep. 410, 335–466 (2005).

    Article  Google Scholar 

  5. 5.

    A.W. Steiner, M. Prakash, J.M. Lattimer et al., Isospin asymmetry in nuclei and neutron stars. Phys. Rep. 411, 325–375 (2005).

    Article  Google Scholar 

  6. 6.

    L.W. Chen, C.M. Ko, B.A. Li et al., Probing the nuclear symmetry energy with heavy-ion reactions induced by neutron-rich nuclei. Front. Phys. China 2, 327–357 (2007).

    Article  Google Scholar 

  7. 7.

    B.A. Li, L.W. Chen, C.M. Ko, Recent progress and new challenges in isospin physics with heavy-ion reactions. Phys. Rep. 464, 113–281 (2008).

    Article  Google Scholar 

  8. 8.

    J.B. Natowitz, G. Röpke, S. Typel et al., Symmetry energy of dilute warm nuclear matter. Phys. Rev. Lett. 104, 202501 (2010).

    Article  Google Scholar 

  9. 9.

    B.M. Tsang, J.R. Stone, F. Camera et al., Constraints on the symmetry energy and neutron skins from experiments and theory. Phys. Rev. C 86, 015803 (2012).

    Article  Google Scholar 

  10. 10.

    W. Trautmann, H.H. Wolter, Elliptic flow and the symmetry energy at supra-saturation density. Int. J. Mod. Phys. E 21, 1230003 (2012).

    Article  Google Scholar 

  11. 11.

    L.W. Chen, C.M. Ko, B.A. Li et al., Probing isospin- and momentum-dependent nuclear effective interactions in neutron-rich matter. Eur. Phys. J. A 50, 29 (2014).

    Article  Google Scholar 

  12. 12.

    C.J. Horowitz, E.F. Brown, Y. Kim et al., A way forward in the study of the symmetry energy: experiment, theory, and observation. J. Phys. G 41, 093001 (2014).

    Article  Google Scholar 

  13. 13.

    B.A. Li, A. Ramos, G. Verde et al., Topical issue on nuclear symmetry energy. Eur. Phys. J. A 50, 9 (2014).

    Article  Google Scholar 

  14. 14.

    X.Q. Liu, M.R. Huang, R. Wada et al., Symmetry energy extraction from primary fragments in intermediate heavy-ion collisions. Nucl. Sci. Tech. 26, S20508 (2015).

    Google Scholar 

  15. 15.

    F.F. Duan, X.Q. Liu, W.P. Lin et al., Investigation on symmetry and characteristic properties of the fragmenting source in heavy-ion reactions through reconstructed primary isotope yields. Nucl. Sci. Tech. 27, 131 (2016).

    Article  Google Scholar 

  16. 16.

    M. Baldo, G.F. Burgio, The nuclear symmetry energy. Prog. Part. Nucl. Phys. 91, 203–258 (2016).

    Article  Google Scholar 

  17. 17.

    B.A. Li, Nucl. Phys. News, in press, (2017) [arXiv:1701.03564]

  18. 18.

    N.K. Glendenning, Compact Stars, 2nd edn. (Spinger, New York, 2000)

    Book  MATH  Google Scholar 

  19. 19.

    J.M. Lattimer, M. Prakash, The physics of neutron stars. Science 304, 536–542 (2004).

    Article  Google Scholar 

  20. 20.

    J.M. Lattimer, M. Prakash, Neutron star observations: prognosis for equation of state constraints. Phys. Rep. 442, 109–165 (2007).

    Article  Google Scholar 

  21. 21.

    J.M. Lattimer, The nuclear equation of state and neutron star masses. Annu. Rev. Nucl. Part. Sci. 62, 485–515 (2012).

    Article  Google Scholar 

  22. 22.

    K. Kotake, K. Sato, K. Takahashi, Explosion mechanism, neutrino burst and gravitational wave in core-collapse supernovae. Rep. Prog. Phys. 69, 971–1143 (2006).

    Article  Google Scholar 

  23. 23.

    H-Th Janka, K. Langanke, A. Marek et al., Theory of core-collapse supernovae. Phys. Rep. 442, 38–74 (2007).

    Article  Google Scholar 

  24. 24.

    M. Hempel, T. Fischer, J. Schaffner-Bielich et al., New equations of state in simulations of core-collapse supernovae. Astrophys. J. 748, 70 (2012).

    Article  Google Scholar 

  25. 25.

    M. Meixner, J.P. Olson, G. Mathews, et al., The NDL equation of state for supernova simulations. arXiv:1303.0064, (2013)

  26. 26.

    M. Oertel, M. Hempel, T. Klähn et al., Equations of state for supernovae and compact stars. Rev. Mod. Phys. 89, 015007 (2017).

    Article  Google Scholar 

  27. 27.

    L.W. Chen, B.J. Cai, C.M. Ko et al., Higher-order effects on the incompressibility of isospin asymmetric nuclear matter. Phys. Rev. C 80, 014322 (2009).

    Article  Google Scholar 

  28. 28.

    L.W. Chen, Higher order bulk characteristic parameters of asymmetric nuclear matter. Sci. China Phys. Mech. Astron. 54, s124–s129 (2011).

    Article  Google Scholar 

  29. 29.

    D.H. Youngblood, H.L. Clark, Y.-W. Lui, Incompressibility of nuclear matter from the giant monopole resonance. Phys. Rev. Lett. 82, 691–694 (1999).

    Article  Google Scholar 

  30. 30.

    S. Shlomo, V.M. Kolomietz, G. Colò, Deducing the nuclear-matter incompressibility coefficient from data on isoscalar compression modes. Eur. Phys. J. A 30, 23–30 (2006).

    Article  Google Scholar 

  31. 31.

    G. Colò, Constraints, Limits and extensions for nuclear energy functionals. AIP Conf. Proc. 1128, 59 (2009).

    Article  Google Scholar 

  32. 32.

    J. Piekarewicz, Do we understand the incompressibility of neutron-rich matter? J. Phys. G 37, 064038 (2010).

    Article  Google Scholar 

  33. 33.

    L.W. Chen, J.Z. Gu, Correlations between the nuclear breathing mode energy and properties of asymmetric nuclear matter. J. Phys. G 39, 035104 (2012).

    Article  Google Scholar 

  34. 34.

    L.W. Chen, Recent progress on the determination of the symmetry energy. Nucl. Struct. China 2012, 43–54 (2013). arXiv:1212.0284

    Article  Google Scholar 

  35. 35.

    B.A. Li, L.W. Chen, F.J. Fattoyev et al., Probing nuclear symmetry energy and its imprints on properties of nuclei, nuclear reactions, neutron stars and gravitational waves. J. Phys. Conf. Ser. 413, 012021 (2013).

    Article  Google Scholar 

  36. 36.

    Z. Zhang, L.W. Chen, Constraining the symmetry energy at subsaturation densities using isotope binding energy difference and neutron skin thickness. Phys. Lett. B 726, 234–238 (2013).

    Article  Google Scholar 

  37. 37.

    B.D. Serot and J.D. Walecka, Advances in Nuclear Physics. Vol. 16, J.W. Negele, E. Vogt, Eds., Plenum, New York (1986)

  38. 38.

    B.D. Serot, J.D. Walecka, Recent progress in quantum hadrodynamics. Int. J. Mod. Phys. E 6, 515 (1997).

    Article  Google Scholar 

  39. 39.

    P.-G. Reinhard, The relativistic mean-field description of nuclei and nuclear dynamics. Rep. Prog. Phys. 52, 439–514 (1989).

    Article  Google Scholar 

  40. 40.

    P. Ring, Relativistic mean field theory in finite nuclei. Prog. Part. Nucl. Phys. 37, 193–263 (1996).

    Article  Google Scholar 

  41. 41.

    J. Meng, H. Toki, S.G. Zhou et al., Relativistic continuum Hartree Bogoliubov theory for ground-state properties of exotic nuclei. Prog. Part. Nucl. Phys. 57, 470–563 (2006).

    Article  Google Scholar 

  42. 42.

    Y. Sugahara, H. Toki, Relativistic mean-field theory for unstable nuclei with non-linear \(\sigma \) and \(\omega \) terms. Nucl. Phys. A 579, 557–572 (1994).

    Article  Google Scholar 

  43. 43.

    Z.Z. Ren, Z.Y. Zhu, Y.H. Cai et al., Relativistic mean-field study of Mg isotopes. Phys. Lett. B 380, 241–246 (1996).

    Article  Google Scholar 

  44. 44.

    G.A. Lalazissis, J. König, P. Ring, New parametrization for the Lagrangian density of relativistic mean field theory. Phys. Rev. C 55, 540 (1997).

    Article  Google Scholar 

  45. 45.

    W.H. Long, J. Meng, N. Van Giai et al., New effective interactions in relativistic mean field theory with nonlinear terms and density-dependent meson-nucleon coupling. Phys. Rev. C 69, 034319 (2004).

    Article  Google Scholar 

  46. 46.

    W.Z. Jiang, Z.Z. Ren, T.T. Wang et al., Relativistic mean-field study for Zn isotopes. Eur. Phys. J. A 25, 29–39 (2005).

    Article  Google Scholar 

  47. 47.

    W.Z. Jiang, Effects of the density dependence of the nuclear symmetry energy on the properties of superheavy nuclei. Phys. Rev. C 81, 044306 (2010).

    Article  Google Scholar 

  48. 48.

    F.J. Fattoyev, C.J. Horowitz, J. Piekarewicz et al., Relativistic effective interaction for nuclei, giant resonances, and neutron stars. Phys. Rev. C 82, 055803 (2010).

    Article  Google Scholar 

  49. 49.

    B.K. Agrawal, A. Sulaksono, P.-G. Reinhard, Optimization of relativistic mean field model for finite nuclei to neutron star matter. Nucl. Phys. A 882, 1–20 (2012).

    Article  Google Scholar 

  50. 50.

    F.J. Fattoyev, J. Carvajal, W.G. Newton et al., Constraining the high-density behavior of the nuclear symmetry energy with the tidal polarizability of neutron stars. Phys. Rev. C 87, 015806 (2013).

    Article  Google Scholar 

  51. 51.

    H. Müller, B.D. Serot, Relativistic mean-field theory and the high-density nuclear equation of state. Nucl. Phys. A 606, 508–537 (1996).

    Article  Google Scholar 

  52. 52.

    C.J. Horowitz, J. Piekarewicz, Neutron star structure and the neutron radius of \(^{208}\)Pb. Phys. Rev. Lett. 86, 5647 (2001).

    Article  Google Scholar 

  53. 53.

    C.J. Horowitz, J. Piekarewicz, Neutron radii of \(^{208}\)Pb and neutron stars. Phys. Rev. C 64, 062802 (2001).

    Article  Google Scholar 

  54. 54.

    C.J. Horowitz, J. Piekarewicz, Constraining URCA cooling of neutron stars from the neutron radius of \(^{208}\)Pb. Phys. Rev. C 66, 055803 (2002).

    Article  Google Scholar 

  55. 55.

    B.G. Todd-Rutel, J. Piekarewicz, Neutron-rich nuclei and neutron stars: a new accurately calibrated interaction for the study of neutron-rich matter. Phys. Rev. Lett. 95, 122501 (2005).

    Article  Google Scholar 

  56. 56.

    L.W. Chen, C.M. Ko, B.A. Li, Isospin-dependent properties of asymmetric nuclear matter in relativistic mean field models. Phys. Rev. C 76, 054316 (2007).

    Article  Google Scholar 

  57. 57.

    B.J. Cai, L.W. Chen, Nuclear matter fourth-order symmetry energy in the relativistic mean field models. Phys. Rev. C 85, 024302 (2012).

    Article  Google Scholar 

  58. 58.

    Equation (23) was given in the first version of the present paper, i.e., arXiv:1402.4242v1 [nucl-th], in February, 2014

  59. 59.

    B.J. Cai, L.W. Chen, Lorentz covariant nucleon self-energy decomposition of the nuclear symmetry energy. Phys. Lett. B 711, 104–108 (2012).

    Article  Google Scholar 

  60. 60.

    L.W. Chen, C.M. Ko, B.A. Li et al., Density slope of the nuclear symmetry energy from the neutron skin thickness of heavy nuclei. Phys. Rev. C 82, 024321 (2010).

    Article  Google Scholar 

  61. 61.

    Z. Zhang, L.W. Chen, Constraining the density slope of nuclear symmetry energy at subsaturation densities using electric dipole polarizability in \(^{208}\)Pb. Phys. Rev. C 90, 064317 (2014).

    Article  Google Scholar 

  62. 62.

    L.W. Chen, Nuclear matter symmetry energy and the symmetry energy coefficient in the mass formula. Phys. Rev. C 83, 044308 (2011).

    Article  Google Scholar 

  63. 63.

    J. Carriere, C.J. Horowitz, J. Piekarewicz, Low-mass neutron stars and the equation of state of dense matter. Astrophys. J. 593, 463–471 (2003).

    Article  Google Scholar 

  64. 64.

    J. Xu, L.W. Chen, B.A. Li et al., Locating the inner edge of the neutron star crust using terrestrial nuclear laboratory data. Phys. Rev. C 79, 035802 (2009).

    Article  Google Scholar 

  65. 65.

    J. Xu, L.W. Chen, B.A. Li et al., Nuclear constraints on properties of neutron star crusts. Astrophys. J. 697, 1549–1568 (2009).

    Article  Google Scholar 

  66. 66.

    G. Baym, C. Pethick, P. Sutherland, The ground state of matter at high densities: equation of state and stellar models. Astrophys. J. 170, 299 (1971).

    Article  Google Scholar 

  67. 67.

    K. Iida, K. Sato, Spin-down of neutron stars and compositional transitions in the cold crustal matter. Astrophys. J. 477, 294–312 (1997).

    Article  Google Scholar 

  68. 68.

    J. Antoniadis, P.C.C. Freire, N. Wex et al., A massive pulsar in a compact relativistic binary. Science 340, 1233232 (2013).

    Article  Google Scholar 

  69. 69.

    P. Demorest, T. Pennucci, S. Ransom et al., A two-solar-mass neutron star measured using Shapiro delay. Nature 467, 1081–1083 (2010).

    Article  Google Scholar 

  70. 70.

    M. Farine, J.M. Pearson, F. Tondeur, Nuclear-matter incompressibility from fits of generalized Skyrme force to breathing-mode energies. Nucl. Phys. A 615, 135–161 (1997).

    Article  Google Scholar 

  71. 71.

    A.W. Steiner, J.M. Lattimer, E.F. Brown, The equation of state from observed masses and radii of neutron stars. Astrophys. J. 722, 33–54 (2010).

    Article  Google Scholar 

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Correspondence to Lie-Wen Chen.

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Dedicated to Joseph B. Natowitz in honour of his 80th birthday.

This work was supported in part by the Major State Basic Research Development Program (973 Program) in China (Nos. 2013CB834405 and 2015CB856904), the National Natural Science Foundation of China (Nos. 11625521, 11275125 and 11135011), the Program for Professor of Special Appointment (Eastern Scholar) at Shanghai Institutions of Higher Learning, Key Laboratory for Particle Physics, Astrophysics and Cosmology, Ministry of Education, China, and the Science and Technology Commission of Shanghai Municipality (No. 11DZ2260700).

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Cai, BJ., Chen, LW. Constraints on the skewness coefficient of symmetric nuclear matter within the nonlinear relativistic mean field model. NUCL SCI TECH 28, 185 (2017).

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  • Equation of state of nuclear matter
  • Heavy-ion collisions
  • Neutron stars