properties of Asymmetric nuclear matter within Extended BHF Approach

Wei ZuoWei ZuoInstitute of Modern Physics, Chinese Institute of Modern Physics, Chinese

Academy of Sciences, LanzhouAcademy of Sciences, Lanzhou

Relativistic many-body problems forheavy and superheavy nuclei

Beijing, June 2009

U. Lombardo, I. Bombaci, G. Fiorela A. Lejeune, B. A. Li ,A. Li, Z. H. Li, J. F. Mathiot, H.-J. Schulze, C.W.Shen, L.G.Cao, H. F. Zhang

• Introduction (Motivation)• Theoretical approaches BHF approach, TBF• Results (TBF effects and TBF rearrangement) Bulk Properties: EOS of ANM, Symmetry enery, EOS at finite Tempertature, Liquid-gas phase Transition Single-particle (s.p.) Properties: Neutron and proton s.p. potentials and effective masses Isospin splitting of nucleon mean fields and effective masses • Summary and conclusion

Outline

MotivationsMotivations

EOS of asymmetric nuclear matter, especially High-density EOS of asymmetric nuclear matter, especially High-density behavior of symmetry energy---- New Challenge ! behavior of symmetry energy---- New Challenge ! P. Danielewicz P. Danielewicz et al., et al., Science 298(2002)1592; B.A.Li, PRL88(2002)192701Science 298(2002)1592; B.A.Li, PRL88(2002)192701

• Nuclear PhysicsNuclear Physics 1) The properties of neutron rich nuclei1) The properties of neutron rich nuclei I. Tanihata, NPA 616 (1997) 560; T. Glasmachet I. Tanihata, NPA 616 (1997) 560; T. Glasmachet et al., et al., PLB 395 (1997)PLB 395 (1997) 2) Strong correlation between the neutron skin thinkness and the slope 2) Strong correlation between the neutron skin thinkness and the slope of symmetry energyof symmetry energy 3) Heavy ion collisions 3) Heavy ion collisions B. A. Li B. A. Li et al., et al., Int. J. Mod. Phys. E7 (1998) 147Int. J. Mod. Phys. E7 (1998) 147

• Implications for astrophysicsImplications for astrophysics J.M. Lattimer and M. Prakash, Science 304 (2004) 536; M.Prakash et al., Phys. Rep. M.Prakash et al., Phys. Rep. 280(1997)1; C.J.Pethick, Rev. Mod. Phys. 64(1992)1133; Lect. Notes Phys., 578 (2001)280(1997)1; C.J.Pethick, Rev. Mod. Phys. 64(1992)1133; Lect. Notes Phys., 578 (2001)

1) Sturctures of neutron stars 1) Sturctures of neutron stars EOS of ANM is a basic input of the nutron star structure modelEOS of ANM is a basic input of the nutron star structure model 2) Chemical Compositions of neutron stars 2) Chemical Compositions of neutron stars determined by symmetry energy at high densitiesdetermined by symmetry energy at high densities 3) Cooling of neutron stars3) Cooling of neutron stars Fast cooling via direct URCA processFast cooling via direct URCA process

properties of Asymmetric Nuclear Matter

Effective NN interaction Effective NN interaction in nuclear mediumin nuclear medium

C. Fuchs and H. H. Wolter, EPJA30(2006)5 Dieperink et al., PRC67(2003)064307.

Symmetry energy predicted by various many-body Symmetry energy predicted by various many-body theories theories ---- ---- Extremely Large uncertainty at high densities!

Effective field theory

DBHF

BHFGreens function

Variational

Most recent results from BHF

Z.H. Li, U. Lombardo, H.-J. Schulze, Zuo et al., PRC74(2006)047304

Theoretical Approaches

• Skyrme-Hartree-Fock• Relativistic Mean Field Theory, Relativistic

Hartree-Fock

• Variational Approach• Green’s Function Theory • Brueckner Theory• Dirac-Brueckner Approach• Effective Field Theory

Theoretical Approaches:

1. Brueckner-hartree-Fock Approach 2. Microscopic Three-Body Force

Bethe-Goldstone Theory

• Bethe-Goldstone equation and effective G-matrix

→ Nucleon-nucleon interaction:

★ Two-body interaction : AV18 (isospin dependent)

★ Effective three-body force

→ Pauli operator :

→ Single particle energy :

→ “Auxiliary” potential : continuous choice

);,()()(

),();,(

21 21

212121

Gikk

kkkkQkkvvG

kkNNNN

effNN Vvv 32

2veffV3

2121 11),( knknkkQ

)()2/()( 22 kUmkk

Ak

kkkkGkkknkU ')]'()(['Re)'()('

Confirmation of the hole-line expansion of the EOS under

the contineous chioce (Song,Baldo,Lombardo,et al,PRL(1998))

Brueckner Theory of Nuclear Matter

Microscopic Three-body Forces

N

R ,

,

)(b )(c

N

N

N

N

N

N

N

, , , ,

N

N

,

,

R,

)(a

,

, ,

Z-diagram

• Based on meson exchange approach• Be constructed in a consistent way with the adopted two-

body force---------microscopic TBF !• Grange et.al PRC40(1989)1040

Effective Microscopic Three-body Force

• Effective three-body force effV3

231333213213

23133*

3321213

11,,',','

'1'1''dd4

1,','

rrrrrrrrrW

rrrrrTrrrrrV

n

nn

eff

→ Defect function: (r12)= (r12) – (r12) ★Short-range nucleon correlations (Ladder correlations) ★Evaluated self-consistently at each iteration

Effective TBF ---- Density dependent

Effective TBF ---- Isospin dependent for asymmetric

nuclear matter

EOS of Nuclear Matter

TBF effect on the EOS of asymmetric nuclear matter

The TBF makes the the EOS much stiffer at high densities

β=0, 0.2, 0.4, 0.6, 0.8, 1

W. Zuo, A. Lejeune, U.Lombardo, J.F.Mothiot, NPA706(2002)418

Z-diagram

Full TBF

Saturation Mechanism

(fm-3) EA (MeV) K (MeV)

0.19 –15.0 210

0.26 –18.0 230

Saturation properties:

TBF is necessary for reproducing the empirical saturation property of nuclear matter in a non-relativistic microscopic framework.

Z-diagram

Full TBF

Relativistic effect in Dirac-BHF approach and TBF effect

W. Zuo et al. NPA706(2002)418

The other elementary processes can not be completely neglected especially at high densities

The comparison between the contribution of the 3BF derived from 2-NN exchange component and relativistic effect in DBHF approach

Z diagram 3BF contribution, Provide by Prof. U. Lombardo

Critical temperature for liquid-gas phase transition in warm nuclear matter

Z-diagram

Full TBF

SHF : 14-20 MeV RMT : 14 MeV DBHF: 10 MeV BHF(2BF): 16 MeVBHF(TBF): 13 MeVBHF(Z-d): 11 MeV

A possible explanation of the discrepancy between the

DBHF and BHF predictions

W. Zuo, Z.H.Li,A. Li, U.lombardo, NPA745(2004)34.

W. Zuo, A. Lejeune, U.Lombardo, J.F.Mothiot,

Parabolic law : linear dependence on β2

W. Zuo et al., PRC69(2004)064001

2( , , ) ( , ,0) ( , )A A symE T E T E T

The EOS of ANM is determined by the EOS of SNM and symmetry energy

sym4n p E

W. Zuo, A. Lejeune, U.Lombardo, J.F.Mothiot, NPA706(2002)418

Density dependence of symmetry energyDensity dependence of symmetry energy

W. Zuo et al. PRC 69(2004)064001

TBF effect Thermal effect

Decomposition of the EOS into various ST channels------ symmetric nuclear matter

squqres: SD

Decomposition of the EOS into various ST channels----- asymmetric nuclear matter

Squares: SDSolid: T=0Dash: ST=00Long-dash: ST=10

Dot: T=1Dot-dash:ST=01Double-dot-dash: ST=11

Single Particle Properties in neutron-rich matter

• Isosping splitting of effective mass• TBF rearrangement cobtribution

• neutron and proton s.p. potential • Isovector part : Symmetry potential

sym

( ) ( )( )

2n pU k U k

U k

Isospin splitting of nucleon mean field

W. Zuo, L.G. Gao, B.A. Li et al., Phys. Rev. C72 (2005)014005 .

In neutron rich matter : Up<Un at low momentaUp>Un at high enough momenta

Nuclear Symmetry Potential in Neutron-rich Matter

Isovector parts of neutron and proton s.p. potentials in neutron-rich matter

Comparison to DBHF predictions: Dalen et al., PRL95(05)022302

F. Sammarruca et al., nucl-th/0411053

BHF prediction: Momentum depndence Density dependence Isospin dependence

sym

( ) ( )( )

2n pU k U k

U k

Nuclear Symmetry Potential in Neutron-rich Matter : Lane potential

Predictions of Skyrme-like interactions

Extended BHF prediction :Comparison with empirical Lane potential

Comparison of the microscopic symmetry potential with the phenomenological ones

Our microscopic symmetry potential shows a strongly different density and momentum dependence from the phenomenological ones adopted in the dynamical simulations of HIC.

It is necessary to apply the

microscopic symmetry potential

in the calculations of HIC.

effective mass describes the non locality of the s.p. energy, which makes thelocal part less attractive. Starting from the energy-moment conservation

2

( , )2p p

pE E p

m

The effective mass is defined as:

*1 1

( )p

p

dEm m

m p p dp E p

effective mass is density and momentum dependent:

p ≤ pF m* > 1 (pairing?)

p > pF m* < 1

definition of m*

Neutron-proton effective mass splitting in neutron-rich matter

M*n > M*p

1* d1

dFk

m m U

m p k

neutrons

protons

Skyrme-like interactions:

mp* < mn* or mn* < mp*

B. A. Li et al., PRC69(2004)064602

Comparison to other predictions:

DBHF: mn* > mp*

Dalen et al., PRL95(2005)022302Z. Y. Ma et al., PLB 604 (2004)170F. Sammarruca et al., nucl-th/0411053

Microscopic origin of the isospin splittingMicroscopic origin of the isospin splitting

Neutron-proton effective masses is controlled by the isospin T=0 SD tensor component of the NN interaction

Neutron-proton effective masses is determined by the isospin splitting of k-mass.

( , ) ( ) ( )

( , ) ( ) ( )

nU U Usym

pU U Usym

BHF numerical prediction

Un-Up is linearly dependent on asymmetry in the consideredrange of asymmetry and momentum (energy)

at high energy Usym changes sign

Isospin splitting of effective mass can be extracted

Lane (1962)

( )* *m mMD MDU U En pm mn p

Provide by Prof. U. Lombardo

Isospin OMPcomparison with collisions p-A n-A

Provide by Prof. U. Lombardo

TBF effects on s.p. properties：1. TBF effect via G-matrix directly

eff3

TBF

1( )

2 i jij k A

Vk ij ij n n

n

3. TBF rearrangement

'

( ) ( ') Re ' [ ( ) ( ')] 'BHF Ak

U k n k kk G k k kk

);,()()(

),();,(

21 21

212121

Gikk

kkkkQkkvvG

kkNNNN

2. Ground state correlations

Full s.p. potential：2( ) ( ) ( ) ( )BHF TBFU k U k U k U k

TBF rearrangment effect on s.p. propertiesZuo, Lombardo, Schulze, Li, Phys. Rev. C74 (2006)017304

S.P. Potential ： Ground state correlation and TBF rearrangement effect

TBF rearrangment contributions to the s.p. potentials

S.p. potentials in SNM in three cases: without the TBF; including the TBF effect only via G-matrix; including the full contribution of the TBF

TBF effects on s.p. properties :

1. TBF affects the s.p. properties via G-matrix

2. TBF rearrangement modifications of the s.p. properties

1. The TBF induces a strongly repulsive

and momentum-dependent rearrangement

modification of the neutron and proton s. p.

potentials at high densities and momenta.

2. The TBF rearrangement contribution is

much larger than that via G-matrix above

the Feimi momentum.

3. The TBF rearrangement strongly reduces

the attraction and enhances the

momentum-dependence of the s.p.

potential at high densities and momenta.

TBF rearrangment effect on symmetry potential

1. Negligible at low densities around and

below the Fermi momentum.

2. Enhancement of the repulsion for

neutrons and the attraction for

protons.

3. Modification of the high-momentum

behavior at high

TBF rearrangment effect on neutron and proton effective masses

1. Remarkable reduction of the neutron

and proton effective masses.

2. Suppression of the isospin splitting

in neutron-rich matter at high

densities.

Symmetric nuclear matter

Zuo, Lombardo, Schulze, Li, Phys. Rev. C74 (2006)017304

Implications for neutron stars

• Proton fraction in neutron star matter

• Kaon condensation

Proton fraction in β-stable neutron star matter

A. Lejeune, U.Lombardo, W. Zuo, Phys.Lett. B477(2000)45

Neutron Star Neutron Star StructureStructure

X.R.Zhou et al., PRC69(2004)018801

Kaon condensation in Kaon condensation in neutron starsneutron stars

Variational

BHF + 3BF

RMT

W. Zuo. A. Li, Z.H.Li, U. Lombardo,

PRC70(2004)055802.

Summary

• The TBF provides a repulsive contribution to the EOS and improves remarkably the predicted saturation properties.

• The TBF from the Z-diagram provides the saturation mechanism and gives the main relativistic effect in DBHF approach.

• The empirical parabolic law for the EOS of ANM can be extended

to the highest asymmetry and to the finite-temperature case.

• The TBF leads to a strong enhancement of symmetry energy and the proton fraction in β-stable matter at high density.

• The neutron-proton effective mass splitting is

• The neutron-proton effective mass splitting is determined by the splitting of the k-mass and essentially controlled by the nature of the NN interaction.

• The TBF induces a strongly repulsive and momentum-dependent rearrangement contribution to the s.p. potential at high densities.

m*n > m*p

谢谢 !THANK YOU!