1. Density-dependence of nuclear symmetry energy
许 昌
南京大学物理学院
Collaborators:
任中洲(NJU), 陈列文(SJTU), 李宝安(TAMU)

2. Outline 1. Brief Introduction of Symmetry Energy
2. Theoretical Formulism (Esym and L)
3. Results and Discussion
4. Short Summary

3. 1. Introduction
Nuclear symmetry energy a key issue in both nuclear physics and astrophysics
symmetry energy
Isospin asymmetry 12 12 12
Energy per nucleon in symmetric nuclear matter 18 18 3
Energy per nucleon in asymmetric nuclear matter

4. A. W. Steiner, M. Prakash, J. M. Lattimer and P. J. Ellis, Phys. Rep
(Effective Field Theory)
(QCD)
n/p
π-/π+
Isospin physics
isodiffusion
isotransport in
isocorrelation
Terrestrial Labs
isofractionation
t/3He
K+/K0
isoscaling 4

5. Recent progress:
1 Experimentally, some constraints on Esym at sub-saturation densities (ρ< ρ0) have been obtained recently from analyzing nuclear reaction data.
2 Esym at normal nuclear density is known to be around 30MeV from analyzing nuclear masses and other data.
3 At supra-saturation densities (ρ> ρ0) , however, the situation is much less clear because of the very limited data available.
BUU calculations…

6. 2. Theoretical Formulism
Starting from the Hugenholtz–Van Hove theorem that is a fundamental relation among the Fermi energy, the average energy per particle E and the pressure of the system P at the absolute temperature of zero.
The nucleon single-particle potentials can be expanded as a power series
isoscalar isovector

7. Lane potential:

8. 2. Theoretical Formulism
Comparing the coefficient of each term then gives the symmetry energy of any order

9. Question:
Nuclear Density Functional Theory (DFT) program at the Institute of Nuclear Theory in Seattle (2005) :
Another goal is to understand connections between the symmetry energy and isoscalar and isovector mean fields?
Symmetry energy: Kinetic energy part, isoscalar potential part, isovector potential part (most uncertain)
Xu et. al, Phys. Rev. C 82, (2010); Xu et. al, Nucl.Phys. A 865 (2011) 1
Xu et. al, Phys. Rev. C 81, (2010)

10. BUU: The Momentum dependent Interaction (MDI)

11. The symmetry energy can be characterized by using the value of Esym(ρ0) and the slope parameter L
L: its exact value is particularly important for determining several critical quantities, such as
the size of the neutron skin in heavy nuclei
location of the neutron drip line
core-crust transition density and gravitational binding energy of neutron stars

12. 3.Symmetry energy and its slope at saturation density
Systematics based on world data accumulated since 1969:
Single particle energy levels from pick-up and stripping reaction
Neutron and proton scattering on the same target at about the same energy
Proton scattering on isotopes of the same element
(p,n) charge exchange reactions

13. Constraining the symmetry energy near saturation density using global nucleon optical potentials
C. Xu, B.A. Li and L.W. Chen, PRC 82, (2010).

14. Constraints extracted from data using various models
GOP: global optical potentials (Lane potentials)
C. Xu, B.A. Li and L.W. Chen, PRC 82, (2010)
Iso. Diff & double n/p (ImQMD, 2009),
M. B. Tsang et al., PRL92, (2009).
Iso Diff. (IBUU04, 2005),
L.W. Chen et al., PRL94, (2005)
IAS+LDM (2009),
Danielewicz and J. Lee, NPA818, 36 (2009)
PDR (2010) of 68Ni and 132Sn,
A. Carbone et al., PRC81, (2010).
PDR (2007) in 208Pb
Land/GSI, PRC76, (2007)
SHF+N-skin of Sn isotopes,
L.W. Chen et al., PRC 82, (2010)
Isoscaling (2007),
D.Shetty et al. PRC76, (2007)
DM+N-Skin (2009): M. Centelles et al., PRL102, (2009)
TF+Nucl. Mass (1996), Myers and Swiatecki, NPA601, 141 (1996) 14

15. Symmetry energy and its slope at saturation density

16. Symmetry energy at supra-saturation density
Some indications of a supersoft Esym at high densities have been obtained from analyzing the π+/π− ratio data.
Experiments have now been planned to investigate the high-density behavior of the Esym at the CSR in China, GSI in Germany, MSU in the United States, and RIKEN in Japan.
Possible physical origins of the very uncertain Esym at supra-saturation densities?

17. Effects of the spin-isospin dependent three-body force
U0: relatively well determined
Usym measures the explicit isospin
dependence of the nuclear strong
interaction, namely, if the effective
interactions are the same in the
isosinglet and isotriplet channels,
then the Usym = 0
However, the Usym is very poorly
known especially at high momenta.
Effects of the spin-isospin dependent three-body force
Effects of the in-medium short range tensor force and nucleon correlation

18. Effects of the spin-isospin dependent three-body force
The symmetry energy obtained with different spin dependence x0 and density dependence α in the three-body force (Gogny force)

19. Effects of the in-medium short-range tensor force
The pion and rho
meson exchanges
tensor forces
We use the Brown-Rho Scaling (BRS)
for the in-medium rho meson mass

20. The symmetry energy with different values of the BRS parameter αBR= 0, 0.05, 0.10, 0.15, 0.20 using different values for the tensor correlation parameter.

21. 4. Summary
General expressions are derived for Esym and L by using the HVH theorem.
Esym and L at normal density: extracted values from the global optical potential
The reason why the Esym and L at supra saturation density so uncertain: isospin-dependence of the three-body force, tensor force, nucleon-nucleon correlation.

22. Thanks!

23. References N. M. Hugenholtz and L. Van Hove, Physica 24, 363 (1958)
C. Xu, B. A. Li, L. W. Chen, and C. M. Ko, ArXiv:
K. A. Brueckner and J. Dabrowski, Phys. Rev. 134, B722 (1964).
J. Decharge and D. Gogny, Phys. Rev. C 21, 1568 (1980).
M. L. Ristig, W. J. Louw, and J. W. Clark, Phys. Rev.C 3, 1504 (1971).