Skip to main content
Log in

Application of microscopic transport model in the study of nuclear equation of state from heavy ion collisions at intermediate energies

  • Topical Review
  • Published:
Frontiers of Physics Aims and scope Submit manuscript

Abstract

The equation of state (EOS) of nuclear matter, i.e., the thermodynamic relationship between the binding energy per nucleon, temperature, density, as well as the isospin asymmetry, has been a hot topic in nuclear physics and astrophysics for a long time. The knowledge of the nuclear EOS is essential for studying the properties of nuclei, the structure of neutron stars, the dynamics of heavy ion collision (HIC), as well as neutron star mergers. HIC offers a unique way to create nuclear matter with high density and isospin asymmetry in terrestrial laboratory, but the formed dense nuclear matter exists only for a very short period, one cannot measure the nuclear EOS directly in experiments. Practically, transport models which often incorporate phenomenological potentials as an input are utilized to deduce the EOS from the comparison with the observables measured in laboratory. The ultrarelativistic quantum molecular dynamics (UrQMD) model has been widely employed for investigating HIC from the Fermi energy (40 MeV per nucleon) up to the CERN Large Hadron Collider energies (TeV). With further improvement in the nuclear mean-field potential term, the collision term, and the cluster recognition term of the UrQMD model, the newly measured collective flow and nuclear stopping data of light charged particles by the FOPI Collaboration can be reproduced. In this article we highlight our recent results on the studies of the nuclear EOS and the nuclear symmetry energy with the UrQMD model. New opportunities and challenges in the extraction of the nuclear EOS from transport models and HIC experiments are discussed.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. B. A. Li, L. W. Chen, and C. M. Ko, Recent progress and new challenges in isospin physics with heavy-ion reactions, Phys. Rep. 464(4–6), 113 (2008)

    ADS  Google Scholar 

  2. M. B. Tsang, J. R. Stone, F. Camera, P. Danielewicz, S. Gandolfi, K. Hebeler, C. J. Horowitz, J. Lee, W. G. Lynch, Z. Kohley, R. Lemmon, P. Möller, T. Murakami, S. Riordan, X. Roca-Maza, F. Sammarruca, A. W. Steiner, I. Vidaña, and S. J. Yennello, Constraints on the symmetry energy and neutron skins from experiments and theory, Phys. Rev. C 86(1), 015803 (2012)

    ADS  Google Scholar 

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

    ADS  Google Scholar 

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

    ADS  Google Scholar 

  5. B. A. Li, B. J. Cai, L. W. Chen, and J. Xu, Nucleon effective masses in neutron-rich matter, Prog. Part. Nucl. Rhys. 99, 29 (2018)

    ADS  Google Scholar 

  6. X. Roca-Maza and N. Paar, Nuclear equation of state from ground and collective excited state properties of nuclei, Prog. Part. Nucl. Phys. 101, 96 (2018)

    ADS  Google Scholar 

  7. S. Burrello, M. Colonna, and H. Zheng, The symmetry energy of the nuclear EoS: A study of collective motion and low-energy reaction dynamics in semiclassical approaches, Front. Phys. 7, 53 (2019)

    Google Scholar 

  8. G. Giuliani, H. Zheng, and A. Bonasera, The many facets of the (non-relativistic) nuclear equation of state, Prog. Part. Nucl. Phys. 76, 116 (2014)

    ADS  Google Scholar 

  9. C. W. Ma and Y. G. Ma, Shannon information entropy in heavy-ion collisions, Prog. Part. Nucl. Phys. 99, 120 (2018)

    ADS  Google Scholar 

  10. A. Ono, Dynamics of clusters and fragments in heavy-ion collisions, Prog. Part. Nucl. Phys. 105, 139 (2019)

    ADS  Google Scholar 

  11. J. Xu, Transport approaches for the description of intermediate-energy heavy-ion collisions, Prog. Part. Nucl. Phys. 106, 312 (2019)

    ADS  Google Scholar 

  12. H. Gao, S. K. Ai, Z. J. Cao, B. Zhang, Z. Y. Zhu, A. Li, N. B. Zhang, and A. Bauswein, Relation between gravitational mass and baryonic mass for non-rotating and rapidly rotating neutron stars, Front. Phys. 15(2), 24603 (2020)

    ADS  Google Scholar 

  13. L. W. Chen, C. M. Ko, B. A. Li, C. Xu, and J. Xu, Probing isospin- and momentum-dependent nuclear effective interactions in neutron-rich matter, Eur. Phys. J. A 50(2), 29 (2014)

    ADS  Google Scholar 

  14. J. Xu, L. W. Chen, B. A. Li, and H. R. Ma, Nuclear constraints on properties of neutron star crusts, Astrophys. J. 697(2), 1549 (2009)

    ADS  Google Scholar 

  15. B. J. Cai and L. W. Chen, Nuclear matter fourth-order symmetry energy in the relativistic mean field models, Phys. Rev. C 85(2), 024302 (2012)

    ADS  Google Scholar 

  16. A. W. Steiner, High-density symmetry energy and direct Urca process, Phys. Rev. C 74(4), 045808 (2006)

    ADS  Google Scholar 

  17. J. Pu, Z. Zhang and L. W. Chen, Nuclear matter fourth-order symmetry energy in nonrelativistic mean-field models, Phys. Rev. C 96, 054311 (2017)

    Google Scholar 

  18. Z. W. Liu, Z. Qian, R. Y. Xing, J. R. Niu and B. Y. Sun, Nuclear fourth-order symmetry energy and its effects on neutron star properties in the relativistic Hartree–Fock theory, Phys. Rev. C 97, 2, 025801 (2018)

    ADS  Google Scholar 

  19. S. A. Bass, et al. (UrQMD-Collaboration), Microscopic models for ultrarelativistic heavy ion collisions, Prog. Part. Nucl. Phys. 41, 255 (1998)

    ADS  Google Scholar 

  20. M. Bleicher, E. Zabrodin, C. Spieles, S. A. Bass, C. Ernst, S. Soff, L. Bravina, M. Belkacem, H. Weber, H. Stöcker, and W. Greiner, Relativistic hadron—hadron collisions in the ultra-relativistic quantum molecular dynamics model, J. Phys. G 25(9), 1859 (1999)

    ADS  Google Scholar 

  21. Q. Li, C. Shen, C. Guo, Y. Wang, Z. Li, J. Lukasik, and W. Trautmann, Nonequilibrium dynamics in heavy-ion collisions at low energies available at the GSI Schwerionen Synchrotron, Phys. Rev. C 83(4), 044617 (2011)

    ADS  Google Scholar 

  22. Q. Li, G. Graf, and M. Bleicher, Ultrarelativistic quantum molecular dynamics calculations of two-pion Hanbury–Brown-Twiss correlations in central Pb–Pb collisions at \(\sqrt {{s_{NN}}} = 2.76\) TeV, Phys. Rev. C 85(3), 034908 (2012)

    ADS  Google Scholar 

  23. J. Aichelin, “Quantum” molecular dynamics — a dynamical microscopic n-body approach to investigate fragment formation and the nuclear equation of state in heavy ion collisions, Phys. Rep. 202(5–6), 233 (1991)

    ADS  Google Scholar 

  24. C. Hartnack, R. K. Puri, J. Aichelin, J. Konopka, S. A. Bass, H. Stöcker, and W. Greiner, Modelling the many-body dynamics of heavy ion collisions: Present status and future perspective, Eur. Phys. J. A 1(2), 151 (1998)

    ADS  Google Scholar 

  25. Q. F. Li, Z. X. Li, S. Soff, M. Bleicher, and H. Stöcker, Probing the equation of state with pions, J. Phys. G 32(2), 151 (2006)

    ADS  Google Scholar 

  26. F. S. Zhang, C. Li, L. Zhu and P. Wen, Production cross sections for exotic nuclei with multinucleon transfer reactions, Front. Phys. 13(6), 132113 (2018)

    ADS  Google Scholar 

  27. Y. Zhang and Z. Li, Elliptic flow and system size dependence of transition energies at intermediate energies, Rhys. Rev. C 74(1), 014602 (2006)

    ADS  Google Scholar 

  28. Y. Zhang, Z. Li, and P. Danielewicz, In-medium NN cross sections determined from the nuclear stopping and collective flow in heavy-ion collisions at intermediate energies, Phys. Rev. C 75(3), 034615 (2007)

    ADS  Google Scholar 

  29. Y. Wang, C. Guo, Q. Li, H. Zhang, Z. Li, and W. Trautmann, Collective flow of light particles in Au+Au collisions at intermediate energies, Phys. Rev. C 89(3), 034606 (2014)

    ADS  Google Scholar 

  30. Y. Wang, C. Guo, Q. Li, H. Zhang, Y. Leifels, and W. Trautmann, Constraining the high-density nuclear symmetry energy with the transverse-momentum-dependent elliptic flow, Phys. Rev. C 89, 044603 (2014)

    ADS  Google Scholar 

  31. Y. Du, Y. Wang, Q. Li, and L. Liu, The effect of Lorentz-like force on collective flows of K+ in Au+Au collisions at 1.5 GeV/nucleon, Sci. China Phys. Mech. Astron. 61, 062011 (2018)

    ADS  Google Scholar 

  32. Y. Liu, Y. Wang, Q. Li, and L. Liu, Collective flows of pions in Au+Au collisions at energies 1.0 and 1.5 GeV/nucleon, Phys. Rev. C 97, 034602 (2018)

    ADS  Google Scholar 

  33. G. Q. Li and R. Machleidt, Microscopic calculation of in-medium nucleon—nucleon cross sections, Phys. Rev. C 48(4), 1702 (1993)

    ADS  Google Scholar 

  34. G. Q. Li and R. Machleidt, Microscopic calculation of in-medium proton—proton cross sections, Phys. Rev. C 49(1), 566 (1994)

    ADS  Google Scholar 

  35. F. Sammarruca and P. Krastev, Effective nucleon—nucleon cross sections in symmetric and asymmetric nuclear matter, Phys. Rev. C 73(1), 014001 (2006)

    ADS  Google Scholar 

  36. H. J. Schulze, A. Schnell, G. Ropke, and U. Lombardo, Nucleon—nucleon cross sections in nuclear matter, Phys. Rev. C 55(6), 3006 (1997)

    ADS  Google Scholar 

  37. C. Fuchs, A. Faessler, and M. El-Shabshiry, Off-shell behavior of the in-medium nucleon-nucleon cross section, Phys. Rev. C 64(2), 024003 (2001)

    ADS  Google Scholar 

  38. H. F. Zhang, Z. H. Li, U. Lombardo, P. Y. Luo, F. Sammarruca, and W. Zuo, Nucleon—nucleon cross sections in dense nuclear matter, Phys. Rev. C 76(5), 054001 (2007)

    ADS  Google Scholar 

  39. H. F. Zhang, U. Lombardo, and W. Zuo, Transport parameters in neutron stars from in-medium NN cross sections, Phys. Rev. C 82(1), 015805 (2010)

    ADS  Google Scholar 

  40. W. G. Love and M. A. Franey, Effective nucleon—nucleon interaction for scattering at intermediate energies, Phys. Rev. C 24, 1073 (1981)

    ADS  Google Scholar 

  41. T. Alm, G. Röpke, W. Bauer, F. Daffin, and M. Schmidt, The in-medium nucleon—nucleon cross section and BUU simulations of heavy-ion reactions, Nucl. Phys. A 587(4), 815 (1995)

    ADS  Google Scholar 

  42. G. J. Mao, Z. X. Li, Y. Z. Zhuo, and Z. Q. Yu, Medium effects on the NN inelastic cross section in relativistic heavy-ion collisions, Phys. Lett. B 327(3–4), 183 (1994)

    ADS  Google Scholar 

  43. G. J. Mao, Z. X. Li, Y. Z. Zhuo, Y. Han, and Z. Yu, Study of in-medium NN inelastic cross section from relativistic Boltzmann-Uehling-Uhlenbeck approach, Phys. Rev. C 49(6), 3137 (1994)

    ADS  Google Scholar 

  44. Q. F. Li, Z. X. Li, and G. J. Mao, Isospin dependence of nucleon-nucleon elastic cross section, Phys. Rev. C 62(1), 014606 (2000)

    ADS  Google Scholar 

  45. Q. F. Li, Z. X. Li, and E. G. Zhao, Density and temperature dependence of nucleon—nucleon elastic cross section, Phys. Rev. C 69(1), 017601 (2004)

    ADS  Google Scholar 

  46. Q. F. Li and Z. X. Li, The isospin dependent nucleon—nucleon inelastic cross section in the nuclear medium, Phys. Lett. B 773, 557 (2017)

    ADS  Google Scholar 

  47. G. D. Westfall, W. Bauer, D. Craig, M. Cronqvist, E. Gaultieri, S. Hannuschke, D. Klakow, T. Li, T. Reposeur, A. M. Vander Molen, W. K. Wilson, J. S. Winfield, J. Yee, S. J. Yennello, R. Lacey, A. Elmaani, J. Lauret, A. Nadasen, and E. Norbeck, Mass dependence of the disappearance of flow in nuclear collisions, Phys. Rev. Lett. 71(13), 1986 (1993)

    ADS  Google Scholar 

  48. D. D. S. Coupland, W. G. Lynch, M. B. Tsang, P. Danielewicz, and Y. X. Zhang, Influence of transport variables on isospin transport ratios, Phys. Rev. C 84(5), 054603 (2011)

    ADS  Google Scholar 

  49. B. A. Li, P. Danielewicz, and W. G. Lynch, Probing the isospin dependence of the in-medium nucleon—nucleon cross sections with radioactive beams, Phys. Rev. C 71(5), 054603 (2005)

    ADS  Google Scholar 

  50. B. A. Li and L. W. Chen, Nucleon—nucleon cross sections in neutron-rich matter and isospin transport in heavy-ion reactions at intermediate energies, Phys. Rev. C 72(6), 064611 (2005)

    ADS  Google Scholar 

  51. Z. Q. Feng, Nuclear in-medium effects and collective flows in heavy-ion collisions at intermediate energies, Phys. Rev. C 85(1), 014604 (2012)

    ADS  Google Scholar 

  52. W. M. Guo, G. C. Yong, Y. Wang, Q. Li, H. Zhang, and W. Zuo, Model dependence of isospin sensitive observables at high densities, Phys. Lett. B 726(1–3), 211 (2013)

    ADS  Google Scholar 

  53. S. A. Bass, C. Hartnack, H. Stöcker, and W. Greiner, Azimuthal correlations of pions in relativistic heavy-ion collisions at 1 GeV/nucleon, Phys. Rev. C 51(6), 3343 (1995)

    ADS  Google Scholar 

  54. Y. Wang, C. Guo, Q. Li, and H. Zhang, The effect of symmetry potential on the balance energy of light particles emitted from mass symmetric heavy-ion collisions with isotopes, isobars and isotones, Sci. China Phys. Mech. Astron. 55(12), 2407 (2012)

    ADS  Google Scholar 

  55. C. Guo, Y. Wang, Q. Li, W. Trautmann, L. Liu, and L. Wu, Influence of the symmetry energy on the balance energy of the directed flow, Sci. China Phys. Mech. Astron. 55(2), 252 (2012)

    ADS  Google Scholar 

  56. P. C. Li, Y. J. Wang, Q. F. Li, and H. F. Zhang, Effects of the in-medium nucleon—nucleon cross section on collective flow and nuclear stopping in heavy-ion collisions in the Fermi-energy domain, Phys. Rev. C 97(4), 044620 (2018)

    ADS  Google Scholar 

  57. P. C. Li, Y. J. Wang, Q. F. Li, and H. F. Zhang, Collective flow and nuclear stopping in heavy ion collisions in Fermi energy domain, Nucl. Sci. Tech. 29(12), 177 (2018)

    Google Scholar 

  58. P. Russotto, P. Z. Wu, M. Zoric, M. Chartier, Y. Leifels, R. C. Lemmon, Q. Li, J. Łukasik, A. Pagano, P. Pawlowski, and W. Trautmann, Symmetry energy from elliptic flow in 197Au+197Au, Phys. Lett. B 697(5), 471 (2011)

    ADS  Google Scholar 

  59. Y. Zhang, Z. Li, C. Zhou, and M. B. Tsang, Effect of isospin-dependent cluster recognition on the observables in heavy ion collisions, Phys. Rev. C 85(5), 051602 (2012)

    ADS  Google Scholar 

  60. K. Zbiri, A. L. Fèvre, J. Aichelin, J. Łukasik, W. Reisdorf, et al., Transition from participant to spectator fragmentation in Au+Au reactions between 60A and 150A MeV, Phys. Rev. C 75(3), 034612 (2007)

    ADS  Google Scholar 

  61. Q. Li, Y. Wang, X. Wang, and C. Shen, Rapidity distribution of protons from the potential version of UrQMD model and the traditional coalescence afterburner, Sci. China Phys. Mech. Astron. 59, 622001 (2016)

    Google Scholar 

  62. W. Reisdorf and H. G. Ritter, Collective flow in heavy-ion collisions, Annu. Rev. Nucl. Part. Sci. 47(1), 663 (1997)

    ADS  Google Scholar 

  63. W. Reisdorf, et al. (FOPI Collaboration), Systematics of azimuthal asymmetries in heavy ion collisions in the regime, Nucl. Phys. A 876, 1 (2012)

    ADS  Google Scholar 

  64. U. Heinz and R. Snellings, Collective flow and viscosity in relativistic heavy-ion collisions, Annu. Rev. Nucl. Part. Sci. 63(1), 123 (2013)

    ADS  Google Scholar 

  65. P. Danielewicz, R. Lacey, and W. G. Lynch, Determination of the equation of state of dense matter, Science 298(5598), 1592 (2002)

    ADS  Google Scholar 

  66. J. Y. Ollitrault, Flow systematics from SIS to SPS energies, Nucl. Phys. A 638(1–2), 195c (1998)

    ADS  Google Scholar 

  67. A. Andronic, J. Lukasik, W. Reisdorf, and W. Trautmann, Systematics of stopping and flow in Au+Au collisions, Eur. Phys. J. A 30(1), 31 (2006)

    ADS  Google Scholar 

  68. A. Andronic, et al. (FOPI Collaboration), Excitation function of elliptic flow in Au+Au collisions and the nuclear matter equation of state, Phys. Lett. B 612(3–4), 173 (2005)

    ADS  Google Scholar 

  69. A. Le Fèvre, Y. Leifels, C. Hartnack, and J. Aichelin, Origin of elliptic flow and its dependence on the equation of state in heavy ion reactions at intermediate energies, Phys. Rev. C 98(3), 034901 (2018)

    ADS  Google Scholar 

  70. Y. M. Zheng, C. M. Ko, B. A. Li, and B. Zhang, Elliptic flow in heavy-ion collisions near the balance energy, Phys. Rev. Lett. 83(13), 2534 (1999)

    ADS  Google Scholar 

  71. D. Persram and C. Gale, Elliptic flow in intermediate energy heavy ion collisions and in-medium effects, Phys. Rev. C 65(6), 064611 (2002)

    ADS  Google Scholar 

  72. T. Gaitanos, C. Fuchs, and H. H. Wolter, Nuclear stopping and flow in heavy-ion collisions and the in-medium NN cross section, Phys. Lett. B 609(3–4), 241 (2005)

    ADS  Google Scholar 

  73. B. A. Li and L. W. Chen, Nucleon—nucleon cross sections in neutron-rich matter and isospin transport in heavy-ion reactions at intermediate energies, Phys. Rev. C 72(6), 064611 (2005)

    ADS  Google Scholar 

  74. M. Kaur and S. Gautam, Influence of the constant and density-dependent scaling of the scattering cross-section on reaction dynamics, J. Phys. G 43, 2, 025103 (2016)

    ADS  Google Scholar 

  75. B. Barker and P. Danielewicz, Shear viscosity from nuclear stopping Phys. Rev. C 99, 034607 (2019)

    ADS  Google Scholar 

  76. Z. Basrak, P. Eudes, and V. de la Mota, Aspects of the momentum dependence of the equation of state and of the residual NN cross section, and their effects on nuclear stopping, Phys. Rev. C 93, 054609 (2016)

    ADS  Google Scholar 

  77. G. Lehaut, et al. (INDRA and ALADIN Collaborations), Study of nuclear stopping in central collisions at intermediate energies, Phys. Rev. Lett. 104(23), 232701 (2010)

    ADS  Google Scholar 

  78. W. Reisdorf, et al. (FOPI Collaboration), Nuclear stopping from 0.09A to 1.93A GeV and its correlation to flow, Phys. Rev. Lett. 92(23), 232301 (2004)

    ADS  Google Scholar 

  79. W. Reisdorf, et al. (FOPI Collaboration), Systematics of central heavy ion collisions in the regime, Nucl. Phys. A 848(3–4), 366 (2010)

    ADS  Google Scholar 

  80. P. Li, Y. Wang, Q. Li, J. Wang, and H. Zhang, Effects of impact parameter filters on observables in heavy-ion collisions at INDRA energies, J. Phys. G 47(3), 035108 (2020)

    ADS  Google Scholar 

  81. Y. Wang, C. Guo, Q. Li, Z. Li, J. Su, and H. Zhang, Influence of differential elastic nucleon—nucleon cross section on stopping and collective flow in heavy-ion collisions at intermediate energies, Phys. Rev. C 94, 024608 (2016)

    ADS  Google Scholar 

  82. J. R. Stone, N. J. Stone, and S. A. Moszkowski, Incom-pressibility in finite nuclei and nuclear matter, Phys. Rev. C 89, 044316 (2014)

    ADS  Google Scholar 

  83. E. Khan and J. Margueron, Determination of the density dependence of the nuclear incompressibility, Phys. Rev. C 88, 034319 (2013)

    ADS  Google Scholar 

  84. E. Khan, J. Margueron, and I. Vidaña, Constraining the nuclear equation of state at subsaturation densities, Phys. Rev. Lett. 109(9), 092501 (2012)

    ADS  Google Scholar 

  85. J. J. Molitoris, D. Hahn, and H. Stöcker, Circumstantial evidence for a stiff nuclear equation of state, Nucl. Phys. A 447, 13 (1986)

    ADS  Google Scholar 

  86. J. J. Molitoris and H. Stöcker, Further evidence for a stiff nuclear equation of state from a transverse-momentum analysis of Ar(1800 MeV/nucleon) + KCl, Phys. Rev. C 32(1), 346 (1985)

    ADS  Google Scholar 

  87. H. Kruse, B. V. Jacak, and H. Stöcker, Microscopic theory of pion production and Sidewards flow in heavy-ion collisions, Phys. Rev. Lett. 54(4), 289 (1985)

    ADS  Google Scholar 

  88. J. Aichelin and C. M. Ko, Subthreshold Kaon production as a probe of the nuclear equation of state, Phys. Rev. Lett. 55(24), 2661 (1985)

    ADS  Google Scholar 

  89. H. Stöcker and W. Greiner, High energy heavy ion collisions — probing the equation of state of highly excited hardronic matter, Phys. Rep. 137(5–6), 277 (1986)

    ADS  Google Scholar 

  90. W. Cassing, V. Metag, U. Mosel, and K. Niita, Production of energetic particles in heavy-ion collisions, Phys. Rep. 188(6), 363 (1990)

    ADS  Google Scholar 

  91. C. Sturm, I. Böttcher, M. D. Bowski, A. Förster, E. Grosse, P. Koczoń, B. Kohlmeyer, F. Laue, M. Mang, L. Naumann, H. Oeschler, F. Pühlhofer, E. Schwab, P. Senger, Y. Shin, J. Speer, H. Ströbele, G. Surówka, F. Uhlig, A. Wagner, and W. Waluś, Evidence for a soft nuclear equation-of-state from Kaon production in heavy-ion collisions, Phys. Rev. Lett. 86(1), 39 (2001)

    ADS  Google Scholar 

  92. C. Fuchs, A. Faessler, E. Zabrodin, and Y. M. Zheng, Probing the nuclear equation of state by K+ production in heavy-ion collisions, Phys. Rev. Lett. 86(10), 1974 (2001)

    ADS  Google Scholar 

  93. C. Hartnack, H. Oeschler, and J. Aichelin, Hadronic matter is soft, Phys. Rev. Lett. 96(1), 012302 (2006)

    ADS  Google Scholar 

  94. Z. Q. Feng, Constraining the high-density behavior of the nuclear equation of state from strangeness production in heavy-ion collisions, Phys. Rev. C 83(6), 067604 (2011)

    ADS  Google Scholar 

  95. A. Le Fèvre, Y. Leifels, W. Reisdorf, J. Aichelin, and C. Hartnack, Constraining the nuclear matter equation of state around twice saturation density, Nucl. Phys. A 945, 112 (2016)

    ADS  Google Scholar 

  96. J. Xu, L.-W. Chen, M. Y. B. Tsang, et al., Understanding transport simulations of heavy-ion collisions at 100A and 400A MeV: Comparison of heavy-ion transport codes under controlled conditions, Phys. Rev. C 93, 044609 (2016)

    ADS  Google Scholar 

  97. M. Dutra, O. Lourenco, J. S. Sa Martins, A. Delfino, J. R. Stone, and P. D. Stevenson, Skyrme interaction and nuclear matter constraints, Phys. Rev. C 85(3), 035201 (2012)

    ADS  Google Scholar 

  98. Y. Wang, C. Guo, Q. Li, A. Le Fèvre, Y. Leifels, and W. Trautmann, Determination of the nuclear incompressibility from the rapidity-dependent elliptic flow in heavy-ion collisions at beam energies 0.4A–1.0A GeV, Phys. Lett. B 778, 207 (2018)

    ADS  Google Scholar 

  99. P. Russotto, et al., Results of the ASY-EOS experiment at GSI: The symmetry energy at suprasaturation density, Phys. Rev. C 94, 034608 (2016)

    ADS  Google Scholar 

  100. L. W. Chen, Higher order bulk characteristic parameters of asymmetric nuclear matter, Sci. China Phys. Mech. Astron. 54(S1), 124 (2011)

    ADS  Google Scholar 

  101. L. W. Chen, B. J. Cai, C. M. Ko, B. A. Li, C. Shen, and J. Xu, Higher-order effects on the incompressibility of isospin asymmetric nuclear matter, Phys. Rev. C 80(1), 014322 (2009)

    ADS  Google Scholar 

  102. B. A. Li, Probing the high density behavior of the nuclear symmetry energy with high energy heavy-ion collisions, Phys. Rev. Lett. 88(19), 192701 (2002)

    ADS  Google Scholar 

  103. Z. Xiao, B. A. Li, L. W. Chen, G. C. Yong, and M. Zhang, Circumstantial evidence for a soft nuclear symmetry energy at suprasaturation densities, Phys. Rev. Lett. 102(6), 062502 (2009)

    ADS  Google Scholar 

  104. Z. Q. Feng and G. M. Jin, Probing high-density behavior of symmetry energy from pion emission in heavy-ion collisions, Phys. Lett. B 683(2–3), 140 (2010)

    ADS  Google Scholar 

  105. N. B. Zhang and B. A. Li, Extracting nuclear symmetry energies at high densities from observations of neutron stars and gravitational waves, Eur. Phys. J. A 55(3), 39 (2019)

    ADS  Google Scholar 

  106. W. J. Xie and B. A. Li, Bayesian inference of high-density nuclear symmetry energy from radii of canonical neutron stars, Astrophys. J. 883(2), 174 (2019)

    ADS  MathSciNet  Google Scholar 

  107. Y. Zhou, L. Chen, and Z. Zhang, Equation of state of dense matter in the multimessenger era, Phys. Rev. D 99(12), 121301 (2019)

    ADS  Google Scholar 

  108. Y. Wang, C. Guo, Q. Li, and H. Zhang, 3H/3He ratio as a probe of the nuclear symmetry energy at sub-saturation densities, Eur. Phys. J. A 51, 37 (2015)

    ADS  Google Scholar 

  109. M. Di Toro, V. Baran, M. Colonna, and V. Greco, Probing the nuclear symmetry energy with heavy-ion collisions, J. Phys. G 37(8), 083101 (2010)

    ADS  Google Scholar 

  110. M. B. Tsang, Y. Zhang, P. Danielewicz, M. Famiano, Z. Li, W. G. Lynch, and A. W. Steiner, Constraints on the density dependence of the symmetry energy, Phys. Rev. Lett. 102, 122701 (2009)

    ADS  Google Scholar 

  111. M. B. Tsang, et al., Constraints on the density dependence of the symmetry energy, Int. J. Mod. Phys. E 19, 1631 (2010)

    ADS  Google Scholar 

  112. X. Lopez, et al. (FOPI Collaboration), Isospin dependence of relative yields of K+ and K0 mesons at 1.528A GeV, Phys. Rev. C 75, 011901(R) (2007)

    ADS  Google Scholar 

  113. M. D. Cozma, Neutron—proton elliptic flow difference as a probe for the high density dependence of the symmetry energy, Phys. Lett. B 700(2), 139 (2011)

    ADS  MathSciNet  Google Scholar 

  114. M. D. Cozma, Y. Leifels, W. Trautmann, Q. Li, and P. Russotto, Toward a model-independent constraint of the high-density dependence of the symmetry energy, Phys. Rev. C 88(4), 044912 (2013)

    ADS  Google Scholar 

  115. S. Kumar, Y. G. Ma, G. Q. Zhang, and C. L. Zhou, Sensitivity of neutron to proton ratio toward the high density behavior of the symmetry energy in heavy-ion collisions, Phys. Rev. C 85(2), 024620 (2012)

    ADS  Google Scholar 

  116. L. Lü, H. Yi, Z. Xiao, M. Shao, S. Zhang, G. Xiao, and N. Xu, Conceptual design of the HIRFL-CSR externaltarget experiment, Sci. China Phys. Mech. Astron. 60, 1, 012021 (2017)

    ADS  Google Scholar 

  117. P. Russotto, M. D. Cozma, A. Fèvre, Y. Leifels, R. Lemmon, Q. Li, J. Lukasik, and W. Trautmann, Flow probe of symmetry energy in relativistic heavy-ion reactions, Eur. Phys. J. A 50(2), 38 (2014)

    ADS  Google Scholar 

  118. W. J. Xie, J. Su, L. Zhu, and F. S. Zhang, Symmetry energy and pion production in the Boltzmann-Langevin approach, Phys. Lett. B 718(4–5), 1510 (2013)

    ADS  Google Scholar 

  119. Q. Li, Z. Li, S. Soff, R. K. Gupta, M. Bleicher, and H. Stöcker, Probing the density dependence of the symmetry potential in intermediate-energy heavy ion collisions, J. Phys. G 31(11), 1359 (2005)

    ADS  Google Scholar 

  120. Q. Li, Z. Li, E. Zhao, and R. K. Gupta, Σ+ ratio as a candidate for probing the density dependence of the symmetry potential at high nuclear densities, Phys. Rev. C 71(5), 054907 (2005)

    ADS  Google Scholar 

  121. S. Gautam, A. D. Sood, R. K. Puri, and J. Aichelin, Isospin effects in the disappearance of flow as a function of colliding geometry, Phys. Rev. C 83(1), 014603 (2011)

    ADS  Google Scholar 

  122. M. B. Tsang, et al. (SRIT Collaboration), Pion production in rare-isotope collisions, Phys. Rev. C 95, 044614 (2017)

    ADS  Google Scholar 

  123. B. A. Li, L. W. Chen, F. J. Fattoyev, W. G. Newton, and C. Xu, 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)

    Google Scholar 

  124. L. W. Chen, Recent progress on the determination of the symmetry energy, arXiv: 1212.0284 [nucl-th](2012)

  125. H. Wolter, Proceedings of Science (Bormio 2012), 059 (2012)

  126. B. A. Li and X. Han, Constraining the neutron-proton effective mass splitting using empirical constraints on the density dependence of nuclear symmetry energy around normal density, Phys. Lett. B 727(1–3), 276 (2013)

    ADS  Google Scholar 

  127. P. Danielewicz and J. Lee, Symmetry energy II: Isobaric analog states, Nucl. Phys. A 922, 1 (2014)

    ADS  Google Scholar 

  128. X. Roca-Maza, M. Brenna, B. K. Agrawal, P. F. Bortignon, G. Colo, L. G. Cao, N. Paar, and D. Vretenar, Giant quadrupole resonances in 208Pb, the nuclear symmetry energy, and the neutron skin thickness, Phys. Rev. C 87(3), 034301 (2013)

    ADS  Google Scholar 

  129. B. A. Brown, Constraints on the skyrme equations of state from properties of doubly magic nuclei, Phys. Rev. Lett. 111(23), 232502 (2013)

    ADS  Google Scholar 

  130. Z. Zhang and L. W. Chen, Constraining the symmetry energy at subsaturation densities using isotope binding energy difference and neutron skin thickness, Phys. Lett. B 726(1–3), 234 (2013)

    ADS  Google Scholar 

  131. N. Wang, M. Liu, L. Ou, and Y. Zhang, Properties of nuclear matter from macroscopic-microscopic mass formulas, Phys. Lett. B 751, 553 (2015)

    ADS  Google Scholar 

  132. X. Fan, J. Dong, and W. Zuo, Density-dependent symmetry energy at subsaturation densities from nuclear mass differences, Phys. Rev. C 89(1), 017305 (2014)

    ADS  Google Scholar 

  133. L. W. Chen, C. M. Ko, and B. A. Li, Light clusters production probe to nuclear symmetry energy, Phys. Rev. C 68(1), 017601 (2003)

    ADS  Google Scholar 

  134. L. W. Chen, C. M. Ko, and B. A. Li, Light cluster production in intermediate energy heavy-ion collisions induced by neutron-rich nuclei, Nucl. Phys. A 729(2–4), 809 (2003)

    ADS  Google Scholar 

  135. L. W. Chen, C. M. Ko, and B. A. Li, Effects of momentum-dependent nuclear potential on two-nucleon correlation functions and light cluster production in intermediate energy heavy-ion collisions, Phys. Rev. C 69(5), 054606 (2004)

    ADS  Google Scholar 

  136. Q. Li, Z. Li, S. Soff, M. Bleicher, and H. Stöcker, Probing the density dependence of the symmetry potential at low and high densities, Phys. Rev. C. 72(3), 034613 (2005)

    ADS  Google Scholar 

  137. Y. Zhang and Z. Li, Probing the density dependence of the symmetry potential with peripheral heavy-ion collisions, Phys. Rev. C 71(2), 024604 (2005)

    ADS  Google Scholar 

  138. G. C. Yong, B. A. Li, L. W. Chen, and X. C. Zhang, Triton-3He relative and differential flows as probes of the nuclear symmetry energy at supra-saturation densities, Phys. Rev. C 80(4), 044608 (2009)

    ADS  Google Scholar 

  139. B. A. Li, Neutron—proton differential flow as a probe of isospin-dependence of the nuclear equation of state, Phys. Rev. Lett. 85(20), 4221 (2000)

    ADS  Google Scholar 

  140. V. Greco, V. Baran, M. Colonna, M. Di Toro, T. Gaitanos, and H. H. Wolter, Relativistic effects in the search for high density symmetry energy, Phys. Lett. B 562(3–4), 215 (2003)

    ADS  Google Scholar 

  141. W. Trautmann, M. Chartier, Y. Leifels, R. C. Lemmon, Q. Li, J. Łukasik, A. Pagano, P. Pawłowski, P. Russotto, and P. Wu, Differential neutron-proton squeeze—out, Prog. Part. Nucl. Phys. 62(2), 425 (2009)

    ADS  Google Scholar 

  142. W. Trautmann, et al., The symmetry energy in nuclear reactions, Int. J. Mod. Phys. E 19, 1653 (2010)

    ADS  Google Scholar 

  143. M. D. Cozma, Feasibility of constraining the curvature parameter of the symmetry energy using elliptic flow data, Eur. Phys. J. A 54, 40 (2018)

    ADS  Google Scholar 

  144. Y. Wang, Q. Li, A. Le Fèvre, and Y. Leifels, Study of the nuclear symmetry energy from the rapidity-dependent elliptic flow in heavy-ion collisions around 1 GeV/nucleon regime, Phys. Lett. B 802, 135249 (2020)

    Google Scholar 

  145. Y. X. Zhang, et al., Comparison of heavy-ion transport simulations: Collision integral in a box, Phys. Rev. C 97, 034625 (2018)

    ADS  Google Scholar 

  146. A. Ono, et al., Comparison of heavy-ion transport simulations: Collision integral with pions and Δ resonances in a box, Phys. Rev. C 100, 044617 (2019)

    ADS  Google Scholar 

  147. Y. F. Guo and G. C. Yong, High pt squeezed-out n/p ratio as a probe of Ksym of the symmetry energy, Phys. Rev. C 100, 014617 (2019)

    ADS  Google Scholar 

  148. Y. F. Guo and G. C. Yong, Effects of curvature of the symmetry energy in Sn+Sn reactions at 270 MeV/nucleon, arXiv: 1909.13566 [nucl-th](2019)

  149. B. A. Li, Symmetry potential of the Δ(1232) resonance and its effects on the π/π+ ratio in heavy-ion collisions near the pion-production threshold, Phys. Rev. C 92, 034603 (2015)

    ADS  Google Scholar 

  150. M. D. Cozma, Constraining the density dependence of the symmetry energy using the multiplicity and average pT ratios of charged pions, Phys. Rev. C 95, 014601 (2017)

    ADS  Google Scholar 

  151. Z. Zhang and C. M. Ko, Medium effects on pion production in heavy ion collisions, Phys. Rev. C 95, 064604 (2017)

    ADS  Google Scholar 

  152. W. M. Guo, G. C. Yong, H. Liu, and W. Zuo, Effects of pion potential and nuclear symmetry energy on the π+ ratio in heavy-ion collisions at beam energies around the pion production threshold, Phys. Rev. C 91, 054616 (2015)

    ADS  Google Scholar 

  153. W. M. Guo, G. C. Yong, and W. Zuo, Effect of Δ potential on the π+ ratio in heavy-ion collisions at intermediate energies, Phys. Rev. C 92, 054619 (2015)

    ADS  Google Scholar 

  154. G. C. Yong, Modeling pion production in heavy-ion collisions at intermediate energies, Phys. Rev. C 96, 044605 (2017)

    ADS  Google Scholar 

  155. T. Song and C. M. Ko, Modifications of the pion-production threshold in the nuclear medium in heavy ion collisions and the nuclear symmetry energy, Phys. Rev. C 91, 014901 (2015)

    ADS  Google Scholar 

  156. Q. Li and Z. Li, The isospin dependent nucleon—nucleon inelastic cross section in the nuclear medium, Phys. Lett. B 773, 557 (2017)

    ADS  Google Scholar 

  157. Q. Li and Z. Li, The density- and isospin-dependent Δ-formation cross section and its decay width, Sci. China Phys. Mech. Astron. 62, 972011 (2019)

    Google Scholar 

  158. Y. Cui, Y. Zhang, and Z. Li, In-medium NN → NΔ cross section and its dependence on effective Lagrange parameters in isospin-asymmetric nuclear matter, Chin. Phys. C 43, 024105 (2019)

    ADS  Google Scholar 

  159. Y. Cui, Y. Zhang, and Z. Li, The Δ mass dependence of the M matrix and its influence on the NΔ → NN cross-sections, Chin. Phys. C 44, 024106 (2020)

    ADS  Google Scholar 

  160. Y. Cui, Y. Zhang, and Z. Li, Effect of energy conservation on the in-medium NN → NΔ cross section in isospin-asymmetric nuclear matter, Phys. Rev. C 98, 054605 (2018)

    ADS  Google Scholar 

  161. W. M. Guo, G. C. Yong, and W. Zuo, Effects of nuclear symmetry energy and in-medium NN cross section in heavy-ion collisions at beam energies below the pion production threshold, Phys. Rev. C 90, 044605 (2014)

    ADS  Google Scholar 

  162. Y. Gao, G. C. Yong, L. Zhang, and W. Zuo, Influence of the nuclear symmetry energy on the collective flows of charged pions, Phys. Rev. C 97, 014609 (2018)

    ADS  Google Scholar 

  163. S. J. Cheng, G. C. Yong, and D. H. Wen, Effects of the symmetry energy in the 132Sn +124 Sn reaction at 300 MeV/nucleon, Phys. Rev. C 94, 064621 (2016)

    ADS  Google Scholar 

  164. B. A. Li and L. W. Chen, Neutron—proton effective mass splitting in neutron-rich matter and its impacts on nuclear reactions, Mod. Phys. Lett. A 30, 1530010 (2015)

    ADS  Google Scholar 

  165. W. J. Xie, J. Su, L. Zhu, and F. S. Zhang, Neutron—proton effective mass splitting in a Boltzmann–Langevin approach, Phys. Rev. C 88, 061601(R) (2013)

    ADS  Google Scholar 

  166. W. J. Xie, Z. Q. Feng, J. Su, and F. S. Zhang, Probing the momentum-dependent symmetry potential via nuclear collective flows, Phys. Rev. C 91, 054609 (2015)

    ADS  Google Scholar 

  167. Z. Q. Feng, Momentum dependence of the symmetry potential and its influence on nuclear reactions, Phys. Rev. C 84(2), 024610 (2011)

    ADS  Google Scholar 

  168. Z. Q. Feng, Nuclear dynamics and particle production near threshold energies in heavy-ion collisions, Nucl. Sci. Tech. 29, 40 (2018)

    Google Scholar 

  169. Z. Q. Feng, Effective mass splitting of neutron and proton and isospin emission in heavy-ion collisions, Nucl. Phys. A 878, 3 (2012)

    ADS  Google Scholar 

  170. V. Giordano, M. Colonna, M. Di Toro, V. Greco, and J. Rizzo, Isospin emission and flow at high baryon density: A test of the symmetry potential, Phys. Rev. C 81(4), 044611 (2010)

    ADS  Google Scholar 

  171. L. Y. Tong, P. C. Li, F. P. Li, Y. J. Wang, Q. F. Li, and F. X. Liu, Effects of the nucleon effective mass splitting and density-dependent symmetry energy on the elliptic flow in heavy ion collisions at Elab=0.09 −1.5 GeV/nucleon, Chin. Phys. C (2020) (accepted)

  172. O. Hen, B. A. Li, W. J. Guo, L. B. Weinstein, and E. Piasetzky, Symmetry energy of nucleonic matter with tensor correlations, Phys. Rev. C 91, 025803 (2015)

    ADS  Google Scholar 

  173. B. A. Li, W. J. Guo, and Z. Shi, Effects of the kinetic symmetry energy reduced by short-range correlations in heavy-ion collisions at intermediate energies, Phys. Rev. C 91, 044601 (2015)

    ADS  Google Scholar 

  174. H. L. Liu, G. C. Yong and D. H. Wen, Probing the momentum dependence of the symmetry potential by the free n/p ratio of pre-equilibrium emission, Phys. Rev. C 91, 024604 (2015)

    ADS  Google Scholar 

  175. G. C. Yong, Constraining nucleon high momentum in nuclei, Phys. Lett. B 765, 104 (2017)

    ADS  Google Scholar 

  176. G. C. Yong and B. A. Li, Interplay of short-range correlations and nuclear symmetry energy in hard-photon production from heavy-ion reactions at Fermi energies, Phys. Rev. C 96, 064614 (2017)

    ADS  Google Scholar 

  177. Z. X. Yang, X. H. Fan, G. C. Yong, and W. Zuo, Effects of the initialization of nucleon momentum in heavy-ion collisions at medium energies, Phys. Rev. C 98, 014623 (2018)

    ADS  Google Scholar 

  178. G. C. Yong, Probing proton transition momentum in neutron-rich matter, Phys. Lett. B 776, 447 (2018)

    ADS  Google Scholar 

  179. Z. X. Yang, X. L. Shang, G. C. Yong, W. Zuo, and Y. Gao, Nucleon momentum distributions in asymmetric nuclear matter, Phys. Rev. C 100, 054325 (2019)

    ADS  Google Scholar 

  180. B. A. Li, P. G. Krastev, D. H. Wen, and N. B. Zhang, Towards understanding astrophysical effects of nuclear symmetry energy, Eur. Phys. J. A 55, 117 (2019)

    ADS  Google Scholar 

  181. M. B. Tsang, W. G. Lynch, P. Danielewicz, and C. Y. Tsang, Symmetry energy constraints from GW170817 and laboratory experiments, Phys. Lett. B 795, 533 (2019)

    ADS  Google Scholar 

  182. L. Baiotti, Gravitational waves from neutron star mergers and their relation to the nuclear equation of state, Prog. Part. Nucl. Phys. 109, 103714 (2019)

    Google Scholar 

Download references

Acknowledgements

We are very grateful to Zhuxia Li, Hongfei Zhang, Chenchen Guo, Arnaud Le Fèvre, Yvonne Leifels, and Wolfgang Trautmann for collaborating with us on some of the topics discussed in this review. The authors acknowledge supports by the computing server C3S2 in Huzhou University. The work was supported in part by the National Natural Science Foundation of China (Nos. 11875125, 11947410, and 11505057), and the Zhejiang Provincial Natural Science Foundation of China under Grant Nos. LY18A050002 and LY19A050001, and the “Ten Thousand Talent Program” of Zhejiang Province (Grant No. 2018R52017).

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Qing-Feng Li.

Additional information

arXiv: 2004.11737.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, YJ., Li, QF. Application of microscopic transport model in the study of nuclear equation of state from heavy ion collisions at intermediate energies. Front. Phys. 15, 44302 (2020). https://doi.org/10.1007/s11467-020-0964-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11467-020-0964-6

Keywords

Navigation