1. Higher-Order Effects on the Incompressibility of Isospin Asymmetric Nuclear Matter Lie-Wen Chen ( 陈列文 ) (Institute of Nuclear, Particle, Astronomy, and Cosmology-INPAC, Department of Physics, Shanghai Jiao Tong University) The International Workshop on Nuclear Dynamics in Heavy-Ion Reactions and the Symmetry Energy (IWND09) August 23 ‐ 25, 2009, Shanghai Collaborators ： Bao-Jun Cai and Chun Shen (SJTU) Che Ming Ko and Jun Xu (TAMU) Bao-An Li (TAMU-Commerce)

2. Outline Motivations Formulism Models Saturation properties of asymmetric nuclear matter Constraining the K sat,2 of asymmetric nuclear matter Summary Main Reference: L.W. Chen, B.J. Cai, C.M. Ko, B.A. Li,C. Shen, and J. Xu, Phys. Rev. C 80, 014322 (2009) [arXiv:0905.4323]

3. I. Motivations 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 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 On Earth!!! In Heaven!!!

4. The incompressibility of ANM is a basic property of ANM, and its isospin dependence carries important information on the density dependence of symmetry energy Incompressibility of ANM Incompressibility of ANM around the saturation density ρ 0 The incompressibility of ANM plays an important role for explosions of supernova (see, e.g., E. Baron, J. Cooperstein, and S. Kahana, PRL55, 126(1985)) Giant Monopole Resonance It is generally believed that the incompressibility of ANM at saturation can be extracted experimentally by measuring the GMR in finite nuclei (see, e.g., J. P. Blaizot, Phys. Rep. 61, 171 (1980))

5. Incompressibility of ANM Incompressibility of SNM 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., S. Shlomo et al.,…… __

6. Incompressibility of ANM Incompressibility of ANM around the saturation density ρ 0 Too stiff! Big error bars!

7. Incompressibility of ANM depending on the mass region of nuclei and the number of parameters used in parametrizing the incompressibility of finite nuclei. Incompressibility of ANM around the saturation density ρ 0

8. Incompressibility of ANM Incompressibility of ANM around the saturation density ρ 0

9. Questions What determine the incompressibility of ANM? What can we know about the incompressibility of ANM from the present nuclear data? Are the higher-order isospin asymmetry/density terms important? Can the high density properties of ANM be predicted based on the information around the saturation density? Is the isospin dependent surface term of the incompressibility of neutron-rich nuclei important?

10. II. Formulism EOS of isospin asymmetric nuclear matter The Nuclear Symmetry Energy The 4 th -order Nuclear Symmetry Energy Parabolic Law of EOS for isospin asymmetric nuclear matter

11. EOS of symmetric nuclear matter Parabolic Approximation of EOS for symmetric nuclear matter II. Formulism

12. The Nuclear Symmetry Energy II. Formulism

13. The 4 th -Order Nuclear Symmetry Energy II. Formulism

14. Characteristic Parameters of asymmetric nuclear matter around the normal nuclear matter density II. Formulism

15. Saturation density of asymmetric nuclear matter Binding energy at the saturation density II. Formulism

16. Incompressibility at the saturation density (At saturation, P=0 Isobaric incompressibility) The above expressions are exact and higher-order terms have no contribution! II. Formulism

17. The K sat,2 of asymmetric nuclear matter II. Formulism

18. 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 …… III. Models

19. 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)

20. III. Models Isospin- and momentum-dependent potential (MDI)

21. III. Models Isospin- and momentum-dependent potential (MDI)

22. III. Models Isospin- and momentum-dependent potential (MDI)

23. III. Models Skyrme-Hartree-Fock approach Standard Skyrme Interaction: _________

24. III. Models Skyrme-Hartree-Fock approach

25. III. Models Skyrme-Hartree-Fock approach

26. III. Models Modified Skyrme-Like (MSL) model

27. III. Models Modified Skyrme-Like (MSL) model All the expressions from the above 3 models are analytical! Especially, the Skyrme force parameters can be expressed analytically by a number of physical quantities via the MSL model!

28. IV. Saturation properties of ANM Characteristic parameters and EOS of Asymmetric Nuclear matter It is very difficult to obtain information on the nuclear matter EOS at higher densities from nuclear properties around normal density which can be extracted from nuclear structure of finite nuclei and nuclear excitation! Heavy-Ion Collisions provide an important tool to study the high density EOS!

29. Characteristic parameters and EOS of Asymmetric Nuclear matter The 4-th order symmetry energy is small! IV. Saturation properties of ANM

30. Saturation properties of Asymmetric Nuclear matter By adjusting only one single parameter y, the MSL model can give good description of the symmetry energy predicted by the MDI interaction The saturation properties depend on the density dependence of the nuclear symmetry energy. IV. Saturation properties of ANM

31. Saturation density of Asymmetric Nuclear matter IV. Saturation properties of ANM More neutron-rich nuclear matter has a smaller saturation density The higher-order terms are only important for extremely neutron-rich nuclear matter

32. Binding energy at the saturation density IV. Saturation properties of ANM More neutron-rich nuclear matter has a smaller binding energy The higher-order terms are only important for extremely neutron-rich nuclear matter with a stiff symmetry energy

33. Incompressibility at the saturation density IV. Saturation properties of ANM More neutron-rich nuclear matter has a smaller incompressibility The higher-order terms are only important for extremely neutron-rich nuclear matter with a stiff symmetry energy

34. V. Constraining the K sat,2 parameter K sat,2,K asy, and K sat,4 The higher-order K sat,4 are only important for very stiff symmetry energies The higher-order J 0 contribution generally cannot be neglected!

35. Correlation between K 0 and J 0 V. Constraining the Ksat,2 parameter The J 0 /K 0 displays a good linear correlation with K 0 K 0 J 0 /K 0

36. Correlation between K sym and L V. Constraining the Ksat,2 parameter The K sym displays a good linear correlation with L L K sym

37. Constraining K sat,2 V. Constraining the Ksat,2 parameter K 0 J 0 /K 0 L K sym Only 5 Skyrme forces in the 63 Skyrme forces used are consistent with all empirical constraint: SKM, Gs,Rs,SKO,SKO*

38. V. Constraining the K sat,2 parameter Isospin surface contribution to the incompressibility of finite nuclei M. Brack and W. Stocker, Nucl. Phys. A388 (1982) 230-242 Compressed semi-infinite nuclear matter Surface tension:

39. V. Constraining the K sat,2 parameter Isospin surface contribution to the incompressibility of finite nuclei -537 -702 -526 -522 K τS : 20-30% contribution Including isospin surface term in the incompressibility of finite nuclei can describe Notre Dame data very well!

40. The higher-order K sat,4 parameter is usually very small compared with the K sat,2 parameter The higher-order contribution from J 0 generally cannot be neglected The K sat,2 can be constrained to be -370±120 MeV from present empirical information based on the MSL model The isospin dependent surface term of the incompressibility of neutron-rich nuclei is important More precise constraint on the symmetry energy even around saturation density still remains a big challenge IV. Summary

41. Thanks ！