1. A Status Review on Symmetry Energy around Saturation Density: Theory
Lie-Wen Chen (陈列文)
(INPAC/Department of Physics, Shanghai Jiao Tong University.
Collaborators：
Wei-Zhou Jiang (SEU)
Che Ming Ko and Jun Xu (TAMU)
Bao-An Li and Chang Xu (TAMU-Commerce)
De-Hua Wen (SCUT)
Zhi-Gang Xiao and Ming Zhang (Tsinghua)
Gao-Chan Yong (IMP)
Xin Wang, Bao-Jun Cai, Rong Chen, Peng-Cheng Chu, Kai-Jia Sun, Zhen Zhang, Hao Zheng (SJTU)
“Topical Workshop on Nuclear Symmetry Energy and Astrophysics”, Xi’an, December 17-19, 2010

2. Outline The nuclear symmetry energy
Theoretical tools of determining the symmetry energy around normal density
(1) Many-body approaches
(2) Transport theory
Symmetry energy around normal density from:
(1) Heavy ion collisions
(2) Nuclear structures
(3) Global nucleon optical potential
Symmetry energy around normal density and:
(1) Nuclear effective interactions
(2) Neutron stars
Summary and outlook
p. 1

3. The nuclear symmetry energy
p. 2

4. The Nuclear Symmetry Energy
Liquid-drop model
(Isospin)
Symmetry energy term
Symmetry energy including surface diffusion effects (ys=Sv/Ss)
Sv ~ the nuclear matter symmetry energy at normal density
W. D. Myers, W.J. Swiatecki, P. Danielewicz, P. Van Isacker, A. E. L. Dieperink,……
p. 3

5. The Nuclear Matter Symmetry Energy
EOS of Isospin Asymmetric Nuclear Matter
(Parabolic law)
Symmetric Nuclear Matter
(relatively well-determined)
Symmetry energy term
(poorly known)
Isospin asymmetry
The Nuclear Symmetry Energy
p. 4

6. Astrophysics and Cosmology
The Symmetry Energy
The multifaceted influence of the nuclear symmetry energy A.W. Steiner, M. Prakash, J.M. Lattimer and P.J. Ellis, Phys. Rep. 411, 325 (2005).
Nuclear Physics
on the Earth
Symmetry Energy
Astrophysics and Cosmology
in Heaven
The symmetry energy is also related to some issues of fundamental physics:
1. The precision tests of the SM through atomic parity violation observables (Sil et al., PRC05)
2. Possible time variation of the gravitational constant (Jofre et al. PRL06; Krastev/Li, PRC07)
3. Non-Newtonian gravity proposed in grand unification theories (Wen/Li/Chen, PRL09)
p. 5

7. QCD phase diagram in 3D: density ,temperature, and isospin

8. Theoretical tools of determining the symmetry energy around normal density (1) Many-body approaches (2) Transport theory
p. 7

9. Nuclear Matter EOS: Many-Body Approaches
The nuclear EOS cannot be measured experimentally, its determination thus depends on theoretical approaches
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
Green’s Function Monte Carlo Calculation
Vlowk + Renormalization Group ……
Effective Field Theory
Density Functional Theory (DFT)
Chiral Perturbation Theory (ChPT)
Phenomenological Approaches
Relativistic mean-field (RMF) theory
Quark Meson Coupling (QMC) Model
Relativistic Hartree-Fock (RHF)
Non-relativistic Hartree-Fock (Skyrme-Hartree-Fock)
Thomas-Fermi (TF) approximations
p. 8

10. Esym: Many-Body Approaches
Z.H. Li et al., PRC74, (2006)
Dieperink et al., PRC68, (2003)
BHF
p. 9 10

11. Esym: Many-Body Approaches
Chen/Ko/Li, PRC72, (2005)
Chen/Ko/Li, PRC76, (2007)
p. 10 11

12. W.Z. Jiang, B.A. Li, and L.W. Chen, PLB653, 184 (2007)
Esym: Many-Body Approaches
RMF with chiral limits
W.Z. Jiang, B.A. Li, and L.W. Chen, PLB653, 184 (2007)
p. 11 12

13. Resonant Fermi Gases with a Large Effective Range (PRL,2005)
Pure Nuetron Matter: Many-Body Approaches
Resonant Fermi Gases with a Large Effective Range (PRL,2005)
J. Piekarewicz, JPG37, (2010)
Chiral 3NF (PRC,2010)
Quantum MC (PRC,2010)
Quantum MC (PRL,2008)
p. 12 13

14. Theoretical tools of determining the symmetry energy around normal density (1) Many-body approaches (2) Transport theory
p. 13

15. Transport Theory Central collisions
Transport Models
Ni + Au, E/A = 45 MeV/A
Transport Models for HIC’s at intermediate energies:
N-body approaches
CMD, QMD,IQMD,IDQMD,
ImQMD,ImIQMD,AMD,FMD
One-body approaches
BUU/VUU, BNV, LV, IBL
Relativistic covariant approaches
RVUU/RBUU,RQMD…
Central collisions
Broad applications of transport models
in astrophysics, plasma physics, electron transport in semiconductor and nanostructures, particle and nuclear physics, ……
p. 14

16. Transport model for HIC’s
Isospin-dependent BUU (IBUU) model
Solve the Boltzmann equation using test particle method
Isospin-dependent initialization
Isospin- (momentum-) dependent mean field potential
Isospin-dependent N-N cross sections
a. Experimental free space N-N cross section σexp
b. In-medium N-N cross section from the Dirac-Brueckner
approach based on Bonn A potential σin-medium
c. Mean-field consistent cross section due to m*
Isospin-dependent Pauli Blocking
EOS
p. 15

17. Transport model: IBUU04
Isospin- and momentum-dependent potential (MDI)
Das/Das Gupta/Gale/Li,
PRC67, (2003)
Chen/Ko/Li,
PRL94, (2005)
Li/Chen,
PRC72, (2005)
p. 16

18. Relativistic VUU/BUU Model
Che Ming Ko, Ulrich Mosel, ……
Relativistic Vlasov Equation
+ Collision Term…
Wigner transform, Dirac + Fields Equation
drift
mean field
Non-relativistic Boltzmann-Uehling-Uhlenbeck
“Lorentz Force”→ Vector Fields
pure relativistic term
Collision term:
p. 17

19. 中能重离子碰撞输运模型
QMD模型
关于QMD的“推导”文献J. Aichelin, Phys. Rep.202, 233 (1991)有详细的讨论，其基本出发点同样是多体薛定谔方程，约化密度矩阵以及对BBGKY系列的截断，但一开始就引入相空间的概念，即对密度矩阵作富里叶变换得到所谓的Wigner密度(或称Wigner表示)．从而直接将多体薛定谔方程表示为类似于经典输运方程的形式，这里我们给出有关的主要公式.
AMD/FMD考虑了波函数的反对称化
p. 18

20. 中能重离子碰撞输运模型
QMD模型 r
p. 19

21. 中能重离子碰撞输运模型
QMD模型
p. 20

22. 中能重离子碰撞输运模型
QMD模型 -
p. 21

23. Symmetry energy around normal density from: (1) Heavy ion collisions (2) Nuclear structures (3) Global nucleon optical potential
p. 22

24. Symmetry energy probes
Promising Probes of the Esym(ρ) (an incomplete list !)
Pigmy/Giant resonances
Nucleon optical potential
B.A. Li/L.W. Chen/C.M. Ko
Phys. Rep. 464, 113(2008)
p. 23

25. Esym: Isospin Diffusion in HIC’s Isospin Diffusion/Transport
______________________________________
How to measure
Isospin Diffusion?
PRL84, 1120 (2000)
A+A,B+B,A+B
X: isospin tracer
p. 24

26. (1) Esym: Isospin Diffusion in HIC’s
Symmetry energy, isospin diffusion, in-medium cross section
Chen/Ko/Li, PRL94, (2005)
Chen/Ko/Li, PRC72, (2005)
Li/ Chen, PRC72, (2005)
Isospin Diffusion Data 
Esym(ρ0)=31.6 MeV
L=88±25 MeV
p. 25

27. Esym: Isoscaling in HIC’s Isoscaling observed in many reactions
M.B. Tsang et al. PRL86, 5023 (2001)
p. 26

28. (2) Esym: Isoscaling in HIC’s Consistent with isospin diffusion data!
Constraining Symmetry Energy by Isocaling: TAMU Data
Shetty/Yennello/Souliotis, PRC75,034602(2007); PRC76, (2007)
Isoscaling Data 
Esym(ρ0)=31.6 MeV
L=65 MeV
Consistent with isospin diffusion data!
p. 27

29. (3) Esym: Isospin diffusion and double n/p ratio in HIC’s
ImQMD: n/p ratios and two isospin diffusion measurements
Tsang/Zhang/Danielewicz/Famiano/Li/Lynch/Steiner, PRL 102, (2009)
ImQMD: Isospin Diffusion and double n/p ratio 
Esym(ρ0)~ MeV?
L= MeV
p. 28

30. Symmetry energy around normal density from: (1) Heavy ion collisions (2) Nuclear structures (3) Global nucleon optical potential
p. 29

31. (4) Esym: Nuclear Mass in Thomas-Fermi Model
Myers/Swiatecki, NPA 601, 141 (1996)
Thomas-Fermi Model analysis of 1654 ground state mass of nuclei with N,Z≥8
Thomas-Fermi Model + Nuclear Mass  Esym(ρ0)= MeV L=49.9 MeV
p. 30

32. Esym: Neutron Skin of Heavy Nuclei
Chen/Ko/Li, PRC72, (2005)
Good linear Correlation: S-L
Oyamatsu et al., NPA634, 3 (1998); Brown, PRL85,5296 (2000); Horowitz/Piekarewicz, PRL86,
5647 (2001); Furnstahl, NPA706, 85 (2002); Yoshida/Sagawa, PRC73, (2006)
p. 31 32

33. (5) Esym: Droplet Model Analysis on Neutron Skin
N-Skin data measured in
antiprotonic atoms
Droplet Model + N-skin  Esym(ρ0)= MeV, L=55 ± 25 MeV
p. 32

34. (6) Esym: Skyrme-HF Analysis on Neutron Skin
Chen/Ko/Li/Xu
PRC82, (2010)
N-Skin data of Sn isotopes
Esym(ρ0)=30 MeV
Neutron skin constraints on
L and Esym(ρ0) are insensitive to
the variations of other
macroscopic quantities.
p. 31

35. Esym: Pygmy Dipole Resonances
Electric dipole strength in atomic nuclei
By Deniz Savran
p. 34

36. (7)、(8) Esym: Pygmy Dipole Resonances
Pygmy Dipole Resonances of 130,132Sn  Esym(ρ0)=32 ± 1.8 MeV L= ± 15 MeV
Pygmy Dipole Resonances of 68Ni and 132Sn 
Esym(ρ0)=32.3 ± 1.3 MeV, L=64.8 ± 15.7 MeV
p. 35

37. (9) Esym: IAS+LDM
Danielewicz/Lee, NPA 818, 36 (2009)
Esym from Isobaric Analog States + Liquid Drop model with surface symmetry energy
IAS+Liquid Drop Model with Surface Esym 
Esym(ρ0)=32.5 ± 1 MeV L=94.5 ± 16.5 MeV
p. 36

38. Esym from Liquid Drop model with surface symmetry energy
(10) Esym: LDM
Liu/Wang/Li/Zhang, PRC (2010), in press [arXiv: ]
Esym from Liquid Drop model with surface symmetry energy
Liquid Drop Model with Surface Esym 
Esym(ρ0)=31.1 ± 1.7 MeV L=66.0 ± 13 MeV
p. 37

39. Symmetry energy around normal density from: (1) Heavy ion collisions (2) Nuclear structures (3) Global nucleon optical potential
p. 38

40. Esym: Global nucleon optical potential
Xu/Li/Chen, PRC82, (2010)
Single particle energy at Fermi surface = particle chemical potential
Energy density
Hugenholtz-Von Hove
(HVH) Theorem
Symmetry potential
(Lane potential)
p. 39

41. Esym: Global nucleon optical potential
Xu/Li/Chen, PRC82, (2010)
=26.11 MeV
p. 40

42. (11) Esym: Global nucleon optical potential
Xu/Li/Chen, PRC82, (2010)
p. 41

43. Esym around normal density
11 constraints on Esym (ρ0) and L from nuclear reactions and structures
Esym(ρ0)= MeV
L= MeV
More accurate data are needed to
obtain more stringent constraints!
p. 42

44. Symmetry energy around normal density and: (1) Nuclear effective interactions (2) Neutron stars
p. 43

45. Symmetry energy and Nuclear Effective Interaction
Chen/Ko/Li, PRC76, (2007)
Chen/Ko/Li, PRC72, (2005)
Esym(ρ0)= 31.5 ±4.5 MeV and L=55 ± 25 MeV: only 55/118
Esym(ρ0)= 31.5 ±4.5 MeV and
L=55 ± 25 MeV: only 8/23
p. 44

46. Symmetry energy around normal density and: (1) Nuclear effective interactions (2) Neutron stars
p. 45

47. The Nuclear Symmetry Energy and Neutron Stars
Lattimer/Prakash, Science 304, 536 (2004)
core-crust
transition
Neutron star has solid crust over liquid core.
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 transition density to calculate the fractional moment of inertia of the crust
Link et al., PRL83,3362 (99)
p. 46

48. Locating the inner edge of neutron star crust
Parabolic Law fails!
Xu/Chen/Li/Ma, PRC79, (2009)
Kazuhiro Oyamatsu, Kei Iida
Phys. Rev. C75 (2007)
Parabolic Approximation has been assumed !!!
pasta
Significantly less than their fiducial values:
ρt= fm-3 and Pt=0.65 MeV/fm3
Xu/Chen/Li/Ma, ApJ 697, 1547 (2009)
p. 47

49. Constraints on M-R relation of NS
Xu/Chen/Li/Ma, ApJ 697, 1547 (2009)
Lattimer
Prakash
(Empirical estimate Link et al., PRL83,3362(99))
(Isospin Diff)
p. 48

50. Summary and outlook
p. 49

51. Summary and Outlook
Analyzing the data from heavy ion collisions, nuclear structures
(mass,neutron skin, PDR…), and nucleon global optical potential
already put important constraints on the symmetry energy around
the saturation density: Esym(ρ0) =31.5±4.5 MeV and L=60±30 MeV
More accurate data are needed to obtain more stringent
constraints!
Symmetry energy is important for understanding the effective
nuclear interactions and many properties of neutron stars.
p. 50

52. 谢 谢！ Thanks!

53. Discussions: Esym around normal density
Are important the higher-order characteristic parameters of
the symmetry energy defined at normal density? What’s the
values of these parameters?
How can we determine the Esym(ρ0) more accurately?
Any more novel observables/approaches to extract the
symmetry energy?
Model independent detrmination?
p. 33

54. Discussions: Esym around normal density
p. 33

55. Discussions: Esym around normal density
p. 33

56. Discussions: Esym around normal density
p. 33

57. Discussions: Esym around normal density
p. 33

58. Discussions: Esym around normal density
L.G. Cao, Z.Y. Ma, CPL, 2008
HIC+N-Skin
Trippa, PRC, 2008
M. Liu et al., PRC, 2010
Esym(ρ0)=30±5 MeV
L=58±18 MeV
p. 33

59. Radioactive Ion Beam Facilities
Cooling Storage Ring (CSR) Facility at HIRFL in China
up to 500 MeV/A for 238U
Radioactive Ion Beam Factory (RIBF) at RIKEN in Japan
FAIR-NUSTAR/GSI in Germany
up to 2 GeV/A for 132Sn
SPIRAL2/GANIL in France
/spiral2
Facility of Rare Isotope Beams (FRIB) in the USA
up to 200(400) MeV/A for 132Sn
The Korean Rare Isotope Accelerator (KoRIA)
up to 250 MeV/A for 132Sn, up to 109 pps ……
p. 13

60. The Skyrme HF Energy Density Functional
Standard Skyrme Interaction:
There are more than
120 sets of Skyrme-
like Interactions in
the literature
_________
Agrawal/Shlomo/Kim Au
PRC72, (2005)
Yoshida/Sagawa
PRC73, (2006)
Chen/Ko/Li/Xu
PRC82, (2010)
9 Skyrme parameters:
9 macroscopic nuclear properties:
p. 29

61. The Skyrme HF Energy Density Functional
Chen/Cai/Ko/Li/Shen/Xu, PRC80, (2009): Modified Skyrme-Like (MSL) Model
Chen/Ko/Li/Xu, arXiv:

62. The Skyrme HF with MSL0
Chen/Ko/Li/Xu, arXiv:

63. Correlations between Nuetron-Skin thickness and macroscopic Nuclear Properties
For heavy nuclei 208Pb and 120Sn:
Δrnp is strongly correlated with L, moderately with Esym(ρ0), a little bit with m*s,0
For medium-heavy nucleus 48Ca:
Δrnp correlation with Esym is much weaker; It further depends on GV and W0
Chen/Ko/Li/Xu
PRC82, (2010)
Important Terms
p. 30

64. Esym at very low densities: Clustering effects
Horowitz and Schwenk,
Nucl. Phys. A 776 (2006) 55
S. Kowalski, et al.,
PRC 75 (2007) 64

65. Esym at very low densities: Clustering effects
J. B. Natowitz et al.,
PRL104, (2010) 65