Higher-Order Effects on the Incompressibility of Isospin Asymmetric Nuclear Matter Lie-Wen Chen ( 陈列文 ) (Institute of Nuclear, Particle, Astronomy, and

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 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



Too stiff!

Big error bars!


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 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


0 0 0 0 0Characteristic paramet ( ),er : ,s ,E K J I

III. Models


sym 0 sym sym symCharacteristic parameters: ( ), , , ,E L K J I

III. Models


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




III. Models



III. Models

Modified Skyrme-Like (MSL) model


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!

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!


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.


Saturation density of Asymmetric Nuclear matter


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


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


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


The J0/K0 displays a good linear correlation with K0K0 J0/K0

0

GMR

240 M V

:

20 eK

Correlation between Ksym and L


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


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*


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:


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 ！

Documents

Higher-Order Effects on the Incompressibility of Isospin Asymmetric Nuclear Matter Lie-Wen Chen ( 陈列文 ) (Institute of Nuclear, Particle, Astronomy, and