1. Probing the Equation of State of Neutron-Rich Matter with Heavy-Ion Reactions Bao-An Li Arkansas State University 1.Equation of State and Symmetry Energy of Neutron-Rich Matter Current status and major issues Importance in astrophysics and nuclear physics 2.A Transport Model for Nuclear Reactions Induced by Radioactive Beams Some details of the IBUU04 model Momentum dependence of the isovector nucleon potential in isospin asymmetric matter 3. Determining the Density Dependence of Nuclear Symmetry Energy At sub-saturation densities: isospin transport in heavy-ion reactions and neutron-skin in 208 Pb At higher densities: reactions at RIA and GSI using high energy radioactive beams 4. Summary Collaborators: L.W. Chen, C.M. Ko, Texas A&M University P. Danielewicz and W.G. Lynch, Michigan State University P. Danielewicz and W.G. Lynch, Michigan State University Andrew W. Steiner, Los Alamos National Laboratory Andrew W. Steiner, Los Alamos National Laboratory G.C. Yong and W. Zuo, Chinese Academy of Science G.C. Yong and W. Zuo, Chinese Academy of Science C.B. Das, C. Gale and S. Das Gupta, McGill University C.B. Das, C. Gale and S. Das Gupta, McGill University

2. K. Oyamatsu, I. Tanihata, Y. Sugahara, K. Sumiyoshi and H. Toki, NPA 634 (1998) 3. Equation of State of Neutron-Rich Matter:N/Z saturation lines (TM1) Isospin asymmetry

3. E sym (ρ) predicted by microscopic many-body theories Symmetry energy (MeV) Density Effective field theory DBHF RMF BHF Greens function Variational A.E. L. Dieperink et al., Phys. Rev. C68 (2003) 064307

4. E sym (ρ) from Hartree-Fock approach using different effective interactions E a sym E b sym B. Cochet, K. Bennaceur, P. Bonche, T. Duguet and J. Meyer, nucl-th/0309012 J. Stone, J.C. Miller, R. Koncewicz, P.D. Stevenson and M.R. Strayer, PRC 68, 034324 (2003). Bao-An Li, PRL 88, 192701 (2002) (where paramaterizations of E a sym and E b sym are given) HF predictions using 90 effective interactions scatter between E a sym and E b sym New Skyrme interactions Amplication around normal density

5. The multifaceted influence of symmetry energy in astrophysics and nuclear physics J.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. (2005). Isospin physics Isospin physics n/p isoscaling isoscaling isotransport isotransport isodiffusion isodiffusion t/ 3 He isofractionation isofractionation K + /K 0 isocorrelation isocorrelation Expanding fireball and gamma-ray burst (GRB) from the superdene neutron star (magnetar) SGR 1806-20 on 12/27/2004. RAO/AUI/NSF π-/π+π-/π+π-/π+π-/π+ In pre-supernova explosion of massive stars is easier with smaller symmetry energy GRB and nucleosynthesis in the expanding fireball after an explosion of a supermassive object depends on the n/p ratio

6. The proton fraction x at ß-equilibrium in proto-neutron stars is determined by Critical points of the direct URCA process APR Akmal et al. A.W. Steiner, M. Prakash, J.M. Lattimer and P.J. Ellis, Phys. Rep. (2005). The critical proton fraction for direct URCA process to happen is X c =1/9 from energy-momentum conservation on the proton Fermi surface Slow cooling: modified URCA: Faster cooling by 4 to 5 orders of magnitude: direct URCA Consequence: long surface thermal emission up to a few million years PSR J0205+6449 in 3C58 was suggested as a candidate

7. Promising Probes of the E sym (ρ) in Nuclear Reactions (an incomplete list !) Most proposed probes in heavy-ion collisions are based on transport model studies

8. Hadronic transport equations: Baryons: Mesons: Simulate solutions of the coupled transport equations using test-particles and Monte Carlo: The evolution of is followed on a 6D lattice An example: (gain) (loss) Mean-field potential for baryons The phase space distribution functions, mean fields and collisions integrals are all isospin dependent

9. Symmetry energy and single nucleon potential used in the IBUU04 transport code for reactions with radioactive beams ρ B.A. Li, C.B. Das, S. Das Gupta and C. Gale, PRC 69, 034614; NPA 735, 563 (2004). soft stiff density HF using a modified Gogny force:

10. Momentum and density dependence of the symmetry potential Lane potential extracted from n/p-nucleus scatterings and (p,n) charge exchange reactions provides a boundary condition at ρ 0 : for E kin < 100 MeV P.E. Hodgson, The Nucleon Optical Model, World Scientific, 1994 G.W. Hoffmann et al., PRL, 29, 227 (1972). Consistent with the Lane potential below 100 MeV momentum Density ρ/ρ 0 δ δ

11. 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; NPA 735, 563 (2004).

12. Nucleon-nucleon cross sections and nuclear stopping power in neutron-rich matter in neutron-rich matter in neutron-rich matter is the reduced mass of the colliding pair NN in medium Effects on the nuclear stopping power and nucleon mean free-path in n-rich 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). 1.In-medium xsections are reduced 2.nn and pp xsections are splitted due to the neutron-proton effective mass slitting in neutron-rich matter due to the neutron-proton effective mass slitting in neutron-rich matter

13. Isospin transport (diffusion) in heavy-ion collisions as a probe of E sym (ρ) at subnormal densities Particle Flux: Isospin diffusion coefficient D I depends on the symmetry potential Isospin Flow: L. Shi and P. Danielewicz, Phys. Rev. C68, 017601 (2003).

14. Momentum-independent Momentum-dependent All having the same E sym (ρ)=32 (ρ/ρ 0 ) 1.1 MSU experiments: R=0.42-0.52 in 124 Sn+ 112 Sn at E beam /A=50 MeV mid-central collisions. With 112 Sn+ 112 Sn and 124 Sn+ 124 Sn as references. Use X= 7 Li/ 7 Be or δ of the projectile residue, etc. M.B. Tsang et al. PRL 92, 062701 (2004) Extract the E sym (ρ) from isospin transport Extract the E sym (ρ) from isospin transport A quantitative measure of the isospin non-equilibrium and stopping power In A+B using any isospin tracer X, F. Rami (FOPI), PRL, 84, 1120 (2000). If complete isospin mixing/ equilibrium SBKD: momentum-independent Soft Bertsch-Kruse-Das Gupta EOS Soft Bertsch-Kruse-Das Gupta EOS MDI: Momentum-Dependent Interaction

15. Comparing momentum-dependent IBUU04 calculations with data on isospin transport from NSCL/MSU Experiments favors: E sym (  )=32 (  /ρ 0 ) 1.1 for ρ<1.2ρ 0 K asy (ρ 0 )~-550 MeV Isobaric incompressibility of asymmetric nuclear matter L.W. Chen, C.M. Ko and B.A. Li, PRL 94, 32701 (2005). Strength of isospin transport Next step: 1. Reduce the error bars of the data and the calculations and the calculations 2. Compare with results using other observables 3. Exam effects of in-medium cross sections

16. Predictions for reactions with high energy radioactive beams at RIA and FAIR/GSI Examples: Isospin fractionation π - yields and π - /π + ratio Besides many other interesting physics, it allows the determination of nuclear equation of state for neutron-rich matter at high densities where it is most uncertain and most important for several key questions in astrophysics.

17. Formation of dense, asymmetric nuclear matter at RIA and GSI Soft E sym Stiff E sym n/p ratio of the high density region

18. Isospin fractionation (distillation): at isospin equilibrium EOS requirement: low (high) density region is more neutron-rich with stiff (soft) symmetry energy Isospin asymmetry of free nucleons density Symmetry enengy ρ0ρ0ρ0ρ0 soft stiff

19. Near-threshold pion production with radioactive beams at RIA and GSI However, pion yields are also sensitive to the symmetric part of the EOS yields are more sensitive to the symmetry energy E sym (ρ) since they are mostly produced in the neutron-rich region ρ density stiffsoft

20. Pion ratio probe of symmetry energy

21. Time evolution of π - /π + ratio in central reactions at RIA and GSI soft stiff From the overlapping n-skins of the colliding nuclei

22. Summary The EOS of n-rich matter, especially the E sym (ρ) is very important for many interesting questions in both astrophysics and nuclear physics Transport models are invaluable tools for studying the isospin-dependence of in-medium nuclear effective interactions and properties of n-rich matter A transport model anaysis of recent isospin transport experiments indicates: E sym (  )=32 (  /ρ 0 ) 1.1 for ρ<1.2ρ 0, and K asy (ρ 0 )~-550 MeV High energy radioactive beams at RIA and GSI will allow us to study the EOS of n-rich matter up to 2ρ 0. Several sensitive probes of the E sym (  ) are proposed.

23. Neutron-proton differential flow as a probe of the symmetry energy: Transverse flow as a probe of the nuclear EOS: pxpxpxpx y symmetry potential is generally repulsive for neutrons and attractive for protons for n and p