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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, Dep
artment of Physics, Shanghai Jiao Tong University)
The International Workshop on Nuclear Dynamics in Heavy-Ion Reactions and the Symmetry Energy (IWND09) Augu
st 23‐25, 2009, Shanghai
Collaborators :Bao-Jun Cai and Chun Shen (SJTU)Che Ming Ko and Jun Xu (TAMU)Bao-An Li (TAMU-Commerce)
Outline
Motivations
Formulism
Models
Saturation properties of asymmetric nuclear matter
Constraining the Ksat,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]
I. Motivations
Density Dependence of the Nuclear Symmetry Energy
HIC’s induced by neutron-rich nuclei (CSR/Lanzhou,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
Many-Body Theory
Transport Theory General Relativity
Nuclear Force
EOS for Asymmetric
Nuclear Matter
On Earth!!! In Heaven!!!
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 ρ0
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
GMR AFrequency f K
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))
Incompressibility of ANM
Incompressibility of SNM around the saturation density ρ0
Giant Monopole Resonance 0
22
0 0 2Incompressibility: K =9 ( )
d E
d
K0=231±5 MeVPRL82, 691 (1999)Recent results:K0=240±20 MeVG. Colo et al., U. Garg et al.,S. Shlomo et al.,……
__
GMR 0Frequency f K
Incompressibility of ANM
Incompressibility of ANM around the saturation density ρ0
Too stiff!
Big error bars!
Incompressibility of ANM
sym :
566 1350 34
159 MeV
K
depending on the mass region of nuclei and the number of parameters used in parametrizing the incompressibility offinite nuclei.
Incompressibility of ANM around the saturation density ρ0
Incompressibility of ANMIncompressibility of ANM around the saturation density ρ0
550 1: 00 MeVK
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?
The result of from triggers a lot of debate:
See, e.g.,
H. Sagawa et al., PRC73, 034327 (2007); J. Piekarewicz, PRC76, 031301(R)
550
(2007)
J. Piekarewicz and
100 MeV Notre Dame
K
M. Centelles, PRC79, 054311 (2009); L.W. Chen et al., PRC80, 014322 (2009)
II. Formulism
sym sym,42 4 6( ) (( ,0( , ) ( ),) ) ( ) /n pE E E E O
EOS of isospin asymmetric nuclear matter
The Nuclear Symmetry Energy
2
sym 2
0
1 ( , )( )
2
EE
The 4th-order Nuclear Symmetry Energy
4
sym,4 4
0
1 ( , )( )
4!
EE
Parabolic Law of EOS for isospin asymmetric nuclear matter
sym
sy
2 4
m
( , ) ( ), ( ) /
( , 1
( ,0)
( ,)
)
( 0
(
))
n pE E
E
O
E
E
E
2 3 4 5 0
0
000
00
0
2( ) ( ),
3)
! 3! 4(
!
K J IEE O
0
0
0
22 00 2
33 00 3
44 00 4
0
0
0
( )9 : Incompressibility of symmetric nuclear matter
( )27 : 3rd-order Incompressibility of symmetric nuclear matter
( )81 : 4th-order Incompressibility o
KE
J
I
E
E
f symmetric nuclear matter
EOS of symmetric nuclear matter
Parabolic Approximation of EOS for symmetric nuclear matter
2 3 0000
00( ) ( ) ( ),
2! 3E E O
K
II. Formulism
The Nuclear Symmetry Energy
sym sysym 0
m s2 3 4 5 0y
0
msym 2! 3! 4!
(( ) ( ),)3
EEK I
L OJ
0
0
0
sym
sy
2sym
0 2
2sym2
0 2
3sy
mm3
0 3
( )3 : Slope parameter of the symmetry energy
( )9 : Curvature parameter of the symmetry energy
( )27 : 3rd-order coefficient of the symmetry ener
L
E
J
E
K
E
0
4sym4
0 4sym
gy
( )81 : 4th-order coefficient of the symmetry energyI
E
II. Formulism
The 4th-Order Nuclear Symmetry Energy
2sym, 3 4 5 0sym, s
4 sym,4 sym,4 0
ym,4sy 44 m
0, 2!
( ) ( ),(3! 4
)3!
EK J I
E OL
0
0
0
sym,4
sym,4
2sym,4
0 2
2sym,42
0 2
3sy
symm,43
4 0 3,
( )3 : Slope parameter of the 4th-order symmetry energy
( )9 : Curvature parameter of the 4th-order symmetry energy
( )27 : 3rd-ord
E
E
E
L
K
J
0
4sym,4
s m4
4y ,4 0
er coefficient of the 4th-order symmetry energy
( )81 : 4th-order coefficient of the 4th-order symmetry energyI
E
II. Formulism
0 0 0
sym sym sym
sym,4 sym,4 sym,4 sym,4
0
0 0
sym 0
sym,4 0
Up to 4th-order, there are totally characteristic parameters defin14
( ), ,
( )
( )
ed at
, ,
,
,
,
, ,
,
,
E K J I
LE K J I
L K J IE
Characteristic Parameters of asymmetric nuclear matter around the normal nuclear matter density
0 0 0
sym 0 sym
There are a lot of studies on the following
( ),
characteristic param
eters:
( ),
5
,
E K
E L K
II. Formulism
Saturation density of asymmetric nuclear matter
Binding energy at the saturation density
II. Formulism
Incompressibility at the saturation density
(At saturation, P=0Isobaric incompressibility)
The above expressions are exact and higher-order terms have no contribution!
II. Formulism
The Ksat,2 of asymmetric nuclear matter
then we have:
If we use the parabolic approximation to the EOS of symmetric nuclear matter, i.e.,
2 3 0000
00( ) ( ) ( ),
2! 3E E O
K
sat,2 0
sat,2
0 0 sym
0sym
0
, , and :, i.e.,
are determined by characteristic parameter
s defin t 4
6
ed a K J L K
JK LK L
K
K
symsat,2 asy6K K L K
which has been used extensively in the iterature to characterize the isospin dependence of
the incompressibility of asymmetric nuclear matter.
II. Formulism
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
III. Models
Isospin- and momentum-dependent potential (MDI)
30
0
0
0
0.16 fm
( ) / 16 MeV
MDI Interaction
( ) 31.6 MeV
211 MeV
*/ 0.6
g( o )
8
G ny
sym
E A
E
K
m m
Chen/Ko/Li, PRL94,032701
(2005)Li/Chen, PRC72, 064611
(2005)
Das/Das Gupta/Gale/Li,
PRC67,034611 (2003)
III. Models
Isospin- and momentum-dependent potential (MDI)
0 0 0 0 0Characteristic paramet ( ),er : ,s ,E K J I
III. Models
Isospin- and momentum-dependent potential (MDI)
sym 0 sym sym symCharacteristic parameters: ( ), , , ,E L K J I
III. Models
Isospin- and momentum-dependent potential (MDI)
sym,4 0 sym,4 sym,4 sym,4 sym,4Characteristic parameters ( ), , , ,: E L K J I
III. Models
Skyrme-Hartree-Fock approach
Standard Skyrme Interaction:
_________
III. Models
Skyrme-Hartree-Fock approach
0 0 0 0 0Characteristic paramet ( ),er : ,s ,E K J I
sym 0 sym sym symCharacteristic parameters: ( ), , , ,E L K J I
III. Models
Skyrme-Hartree-Fock approach
sym,4 0 sym,4 sym,4 sym,4 sym,4Characteristic parameters ( ), , , ,: E L K J I
III. Models
Modified Skyrme-Like (MSL) model
0 0 0 0 0Characteristic paramet ( ),er : ,s ,E K J I
III. Models
Modified Skyrme-Like (MSL) model
sym,4 0 sym,4 sym,4 sym,4 sym,4Characteristic parameters ( ), , , ,: E L K J I
sym 0 sym sym symCharacteristic parameters: ( ), , , ,E L K J I
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!
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!
Characteristic parameters and EOS of Asymmetric Nuclear matter
The 4-th order symmetry energy is small!
IV. Saturation properties of ANM
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 interactionThe saturation properties depend on the density dependence of the nuclear symmetry energy.
IV. Saturation properties of ANM
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
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
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
V. Constraining the Ksat,2 parameter
Ksat,2,Kasy, and Ksat,4
The higher-order Ksat,4 are only important for very stiff symmetry energies The higher-order J0 contribution generally cannot be neglected!
Correlation between K0 and J0
V. Constraining the Ksat,2 parameter
The J0/K0 displays a good linear correlation with K0K0 J0/K0
0
GMR
240 M V
:
20 eK
Correlation between Ksym and L
V. Constraining the Ksat,2 parameter
The Ksym displays a good linear correlation with L
L Ksym
(Essentially consistent with
all constraints so
HIC's:
the lower limit from ImQMD
far)
:
111 MeV:
46 111
the
MeV
Note:
upper limi
46 M
t
e
U0
V
IBU 4
L
Constraining Ksat,2
V. Constraining the Ksat,2 parameter
0
sym 0
*s,0
240 20 MeV
( ) 30 5 MeV
0.8 1
: 46 111 MeV
K
E
m m
L
K0 J0/K0
L Ksym
sat,2 370 120 MeVK
Only 5 Skyrme forces in the 63 Skyrme forces used are consistent with all empirical constraint:SKM, Gs,Rs,SKO,SKO*
V. Constraining the Ksat,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:
V. Constraining the Ksat,2 parameter
112
The difference of incompressibility
as a function of A A AK K K
( 125)isoK
A
Isospin surface contribution to the incompressibility of finite nuclei
112
The difference of incompressibility
as a function of A A AK K K
-537 -702 -526 -522 KτS: 20-30% contribution
1/3iso SK K K A
Including isospin surface term in the incompressibility of finite nuclei can describe Notre Dame data very well!
The higher-order Ksat,4 parameter is usually very small compared with the Ksat,2 parameter
The higher-order contribution from J0 generally cannot be neglected
The Ksat,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
Thanks !
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