EOS of Asymmetric Nuclear Matter

Beijing, Aug. 2005

W. ZuoW. ZuoInstitute of Modern Physics, LanZhou, ChinaInstitute of Modern Physics, LanZhou, China

Collaboration

I. Bombaci Pisa UniversityA. Lejeune IPN, LiegeZ. H. Li, G.C.Lu IMP, LanzhouU. Lombardo INFN-LNS, CataniaJ. F. Mathiot Blaise-Pascal Uni.H.-J. Schulze INFN, CataniaC.W.Shen, L.G.Cao INFN-LNS, CataniaB. A. Li Arkansa State University

• Introduction (Motivation)• Theoretical approaches BHF approach, TBF• Results Symmetry enery, EOS at finite Tempertature, TBF effects• Summary

Outline

MotivationsMotivations

• EOS of asymmetric nuclear matter, especially EOS of asymmetric nuclear matter, especially High-density behavior of symmetry energyHigh-density behavior of symmetry energy ---- New Challenge ! ---- 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 M. Di Toro, Phys.Rep. to appearM. Di Toro, Phys.Rep. to appear

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 PLB 395 (1997)(1997) 2) Strong correlation between the neutron skin thinkness 2) Strong correlation between the neutron skin thinkness and the slope of symmetry energyand the slope of 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

MotivationsMotivations

• Implications for astrophysics Implications for astrophysics M.Prakash et al., Phys. Rep. 280(1997)1; M.Prakash et al., Phys. Rep. 280(1997)1; C.J.Pethick, Rev. Mod. Phys. 64(1992)1133;C.J.Pethick, Rev. Mod. Phys. 64(1992)1133; Lect. Notes Phys., 578 (2001)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 EOS of ANM is a basic input of the nutron star structure modelstructure model 2) Chemical Compositions of neutron stars 2) Chemical Compositions of neutron stars determined by symmetry energydetermined by symmetry energy 3) Cooling of neutron stars3) Cooling of neutron stars Fast cooling via direct URCA processFast cooling via direct URCA process

Oyamatsu et al., NPA634(1998)3.

Properties of Neutron-rich NucleiProperties of Neutron-rich Nuclei

R.J.Furnstahl, NPA706(2002)85.

Correlation between symmetry energy and neutron skin thinknessCorrelation between symmetry energy and neutron skin thinkness

B.A.Li, PRL88(2002)192701.

Heavy ion collisionsHeavy ion collisions

J.M. Lattimer and M. Prakash, Science Vol. 304 (2004) 536-542.

Matter in neutron starsMatter in neutron stars

Lattimer et al., PRL66(1991)2701.

Composition of neutron star matter (n,p,e,Composition of neutron star matter (n,p,e,μμ))

=4 4 (1 2 )n p sym symE E x

Lattimer and Prakash, Science 304(2004)536.

Cooling of neutron starsCooling of neutron stars

0.11 0.148cx x Condition for dURCA：

Proton fraction is determined by symmetry energy

Momentum conservation

e p nF F Fp p p

Neutron Star StructureNeutron Star Structure

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

TOV equation

Theoretical Approaches Skyrme-Hartree-FockRelativistic Mean Field Theory, Relativistic

Hartree-Fock

Variational ApproachBrueckner-Hartree-Fock ApproachDirac-Brueckner ApproachEffective Field Theory

B.A.Brown, PRL85(2000)5296

Theoretical predictions of symmetry energyTheoretical predictions of symmetry energy

各种理论模型预言的对称能的密度依赖存在很大的分歧！

Greco et al., PRC63(2001)035202

Theroetical predictions of symmetry energyTheroetical predictions of symmetry energy

Wiringa et al., PRC38(1988)1010.

Dieperink et al., PRC67(2003)064307.

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)'()('

Microscopic Three-body Forces 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

N

R ,

,

)(b )(c

N

N

N

N

N

N

N

, , , ,

N

N

,

,

R,

)(a

,

, ,

Z-diagram

Schematic Comparison between Dirac-BHF & the Microscopic TBF

Leading relativistic correct in Dirac BHF approach

Brown et. al., Comments Nucl. Phys. 17(1987)39; Serot and Walecka, Int. J Mod. Phys. E6(1997)515.

TBF

Contribution of the

TBF to energy per nucleon

Meson parameters :

NN2

N

NN

RU

N N RE

N

NN

NN2

Sv

Sv

) NM Pure ()/(2.6

) SMN ()/(9.3

)(

3/80

3/80

23

23

NNNN VVE

) MeV/400 Provided ( 0SU

NN2

) NM Pure ()/(7.6

) SMN ()/(2.43/8

0

3/80

cmg N GeV/1.1,MeV540,9.114/2

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

TBF effect on the EOS of asymmetric nuclear matter

The TBF makes the the EOS much stiffer at high densities

Asymmetric nuclear matter at finite temperature

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

T=0,8,10,12,14,16MeV

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

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

Z-diagram

Full TBF

Saturation Mechanism

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

0.19 –15.0 210

0.26 –18.0 230

饱和点性质 :

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

Z-diagram

Full TBF

Relativistic effect in Dirac-BHF approach and TBF effect

Z-diagram

Critical temperature for liquid-gas phase transition

Z-diagram

Full TBF

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

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

Parabolic law

W. Zuo, Z.H.Li,A. Li, G.C.Lu, PRC 69(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

TBF effect on symmetry energy

W. Zuo, Z.H.Li,A. Li, G.C.Lu, PRC 69(2004)064001

Isospin splitting of nucleon mean field

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

Neutron-proton effective mass splitting in neutron-rich matter

M*n > M*p

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

1* d1

dFk

m m U

m p k

neutrons

protons

DBHF: mn* > mp* Z. Y. Ma et al., PLB 604 (2004)170

Skyrme-like interactions: mp* < mn* or mn* < mp*

Isosping splitting of k-mass and e-mass

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

neutrons

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

Microscopic origin of the isospin splitting

Neutron-proton effective masses is controlled by the tensor component of the NN interaction

Proton fraction in β-stable neutron star matter

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

Kaon condensation in neutron starsKaon condensation in neutron stars

Variational

BHF + 3BF

RMT

W. Zuo. A. Li, Z.H.Li, U. Lombardo, PRC70(2004)055802.

e Km Critical conditionCritical condition

Kaon Condensed Phase Kaon Condensed Phase

TBF effect on the 1S0 neutron and proton gap in neutron star matter

W. Zuo et al., Phys.Lett. B595(2004)44

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.

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

The TBF suppresses strongly the proton superfluidity in the 1S0 channel induced by the two-body NN interaction.

m*n > m*p

Thank you !

Superfluidity in β-stable matter

The superfludity in a homogeneous Fermi system is discribed by the pairing gap which is determined by the standard BCS gap equation

'' '

1( , ')

2NNk kk k

v k kE

→ Realistic Nucleon-nucleon interaction:

→ Energy spectrum:

→ Single-particle energy :

NNv2 2( )k Fk k

E

2 2 /(2 ) ( )k k m U k

Two main effects are missing from the BCS approach

• screening of the pairing interaction due to the surrounding nucleons (polarization effect)

• medium corrections of the single-particle spectrum

Up to now all investigations have predicted a reduction of the superfluidity gap in the channel due to the above effects..

D.J.Dean, M. H. Jensen, Rev. Mod. Phys. 75(2003)607U.lombardo, H.J.Schulze, Lecture Notes in Physics, vol. 578, 2001.

10S

Screening pairing suppression

U. Lombardo, P. Schuck, W. Zuo, PRC64 (2001) 021301R