The Nuclear Symmetry Energy and Neutron Skin Thickness of Finite Nuclei

Lie-Wen Chen ( 陈列文 )(INPAC and Department of Physics, Shanghai Jiao Tong Un

iversity. lwchen@sjtu.edu.cn)

第十三届全国核结构研讨会暨第九次全国“核结构与量子力学”专题讨论会 ,2010 年 7 月 24-30 日 , 赤峰 , 内蒙古

Collaborators ：Che Ming Ko and Jun Xu (TAMU)Bao-An Li (TAMU-Commerce) Xin Wang (SJTU)Bao-Jun Cai, Rong Chen, Peng-Cheng Chu, Zhen Zhang (SJTU)

Outline

EOS of asymmetric nuclear matter and the symmetry energy

Constraints on density dependence of symmetry energy from nuclear structure and reactions – Present status

Constraining the symmetry energy with the neutron skin thickness of heavy nuclei in a novel correlation analysis

Symmetry energy and nuclear effective interactions

Summary and outlook

Main References: B.A. Li, L.W. Chen, and C.M. Ko, Phys. Rep. 464, 113-281 (2008)L.W. Chen, B.J. Cai, C.M. Ko, B.A. Li, C. Shen, and J. Xu, PRC80, 014322 (200

9)L.W. Chen, C.M. Ko, B.A. Li and J. Xu, arXiv:1004.4672, 2010

Density Dependence of the Nuclear Symmetry Energy

Reactions & Structures of Neutron-Rich Nuclei (CSR/Lanzh

ou, FRIB, GSI,

RIKEN……)

Most uncertain property of an asymmetric

nuclear matter

Isospin Nuclear PhysicsWhat is the isospin dependence of the in-medium nuclear effective interac

tions???

Neutron Stars …

Structures of Neutron-rich Nuclei, …

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!!!

Isospin in Nuclear Physics

EOS of asymmetric nuclear matter and the symmetry energy

EOS of Nuclear Matter

-

The in a nuclear matter with density , temperature , and

isospin asymmetry

T

energ

( ) can be expressed as

T

/

y of per nucleon

( , Nuclear Ma, ) ( tter EO )S

he

n p

E A T

2

, constant

pressure of the nuclear matter can be expresse P

incompessibi

d as

The of the nucl

( , , )

ear mattlty K er can be expressed a

s

T N

P T

, const

300 0

ant

Empirical values about the nuclear mat

Saturation density ( ( ) 0

(

of symmetric nuclear matter a

, ,

t ) T=

ene

ter EOS:

) 9

0.16 fm0 MeV:

The rg

T N

PK T

P

0

0

0

16 MeV/nu

y of per nucleon

Incompessibilty

of symmetric nuclear matter at and T=0 MeV:

of symmetric nuclear matter at T=0 MeV:

cleon

200 400 MeVK

Liquid-drop model

(Isospin) Symmetry energy term

W. D. Myers, W.J. Swiatecki, P. Danielewicz, P. Van Isacker, A. E. L. Dieperink,……

Symmetry energy including surface diffusion effects (ys=Sv/Ss)

The Nuclear Symmetry Energy

EOS of Isospin Asymmetric Nuclear Matters

2 4ym ( )( , ) ( ), ( ),0) /( n pE OE E

(Parabolic law)

The Nuclear Symmetry Energy2

sym 2

1 ( , )( )

2

EE

The Nuclear Matter Symmetry Energy

Symmetry energy term(poorly known)

Symmetric Nuclear Matter(relatively well-determined)

0

sym

sy

2

0 0

0 0

s

sy

0

msym 0

0

0

y

m

m

, ( )3 18

30 MeV (LD mass formula: )

( )3 (Many-Body Theory:

( )

: ; Exp: ???

( )

50 200 e )M

( )

V

E Myers & Swiatecki, NPA81; Pomorski & Du

EL

KL

dek, PRC67

L

K

E E

0

sy

2sym2

0sy

m 0

m ym

0

s2

(Sharma et

isobaric incompressibli

( )9 (Many-Body Theory: : ; Exp: ???)

The isospin part of the of

700 466 Me

6

t

V

320 180

asymmetric nuclear matter

( : / GMR MeV

y K

EK

K K L J K L

Shlomo&Youngblood,PRC47,529(93);

al., PRC38, 2562 (88));

566 1 350 ( 34 159 MeV (T. Li et al, PRL99,162503(2007))550 100 Me )V

The Symmetry Energy

The multifaceted influence of the nuclear symmetry energy A.W. Steiner, M. Prakash, J.M. Lattimer and P.J. Ellis, Phys. Rep. 411, 325 (2005).

The symmetry energy is also related to some issues of fundamental physics:1. The precision tests of the SM through atomic parity violation observables (Sil et al., PRC05)2. Possible time variation of the gravitational constant (Jofre etal. PRL06; Krastev/Li, PRC07)3. Non-Newtonian gravity proposed in grand unification theories (Wen/Li/Chen, PRL10)

Nuclear Matter EOS: Many-Body Approaches

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 ……

Chen/Ko/Li, PRC72, 064309(2005) Chen/Ko/Li, PRC76, 054316(2007)

Z.H. Li et al., PRC74, 047304(2006) Dieperink et al., PRC68, 064307(2003)

BHF

Esym: Many-Body Approaches

At sub-saturation densities (亚饱和密度行为) Sizes of n-skins of unstable nuclei from total reaction cross sections Proton-nucleus elastic scattering in inverse kinematics Parity violating electron scattering studies of the n-skin in 208Pb n/p ratio of FAST, pre-equilibrium nucleons Isospin fractionation and isoscaling in nuclear multifragmentation Isospin diffusion/transport Neutron-proton differential flow Neutron-proton correlation functions at low relative momenta t/3He ratio Hard photon production

Towards high densities reachable at CSR/Lanzhou, FAIR/GSI, RIKEN, GANIL and, FRIB/MSU (高密度行为) π -/π + ratio, K+/K0 ratio? Neutron-proton differential transverse flow n/p ratio at mid-rapidity Nucleon elliptical flow at high transverse momenta n/p ratio of squeeze-out emission

Promising Probes of the Esym(ρ)(an incomplete list !)

Symmetry energy around saturation density

Li/ Chen, PRC72, 064611(2005)

Symmetry energy, isospin diffusion, in-medium cross section

Isospin Diffusion Data Esym(ρ0)=31.6 MeVL=88±25 MeV

0

0

( ) 31.6( / ) MeV

(From 0 ), we ob

Fit the symmetry ener

tain

0.69 for1. 005

gy with

for 1 n

:

a d

symE

x x

Chen/Ko/Li, PRC72,064309 (2005)

Esym: Isospin Diffusion in HIC’s

Chen/Ko/Li, PRL94,032701 (2005)Isospin dependent BUU transport model

Consistent with isospin diffusion data!

Constraining Symmetry Energy by Isocaling: TAMU DataShetty/Yennello/ Souliotis, PRC75,034602(2007); PRC76, 024606 (2007)

Esym: Isoscaling in HIC’s

Isoscaling Data Esym(ρ0)=31.6 MeVL=65 MeV

ImQMD: n/p ratios and two isospin diffusion measurementsTsang/Zhang/Danielewicz/Famiano/Li/Lynch/Steiner, PRL 102, 122701 (2009)

Esym: Isospin diffusion and double n/p ratio in HIC’s

ImQMD: Isospin Diffusion and double n/p ratio Esym(ρ0)=28 - 34 MeVL=38 - 103 MeV

Esym: Nuclear Mass in Thomas-Fermi Model

Thomas-Fermi Model + Nuclear Mass Esym(ρ0)=32 .65 MeV L=49.9 MeV

Myers/Swiatecki, NPA 601, 141 (1996)Thomas-Fermi Model analysis of 1654 ground state mass of nuclei with N,Z≥8

Esym: Pygmy Dipole Resonances

Pygmy Dipole Resonances of 130,132Sn Esym(ρ0)=32 ± 1.8 MeV L=43.125 ± 15 MeV

Pygmy Dipole Resonances of 68Ni and 132Sn Esym(ρ0)=32.3 ± 1.3 MeV, L=64.8 ± 15.7 MeV

Esym from Isobaric Analog States + Liquid Drop model with surface symmetry energy

IAS+Liquid Drop Model with Surface Esym Esym(ρ0)=32.5 ± 1 MeV L=94.5 ± 16.5 MeV

Danielewicz/Lee, NPA 818, 36 (2009)

Esym: IAS+LDM

Esym: Droplet Model Analysis on Neutron Skin

Droplet Model + N-skin Esym(ρ0)=31.6 MeV, L=66.5 ± 36.5 MeV

Droplet Model + N-skin Esym(ρ0)=28 - 35 MeV, L=55 ± 25 MeV

Esym: Droplet Model Analysis on Neutron Skin

Esym around normal density

Esym(ρ0)=28 - 35 MeVL=28 - 111 MeV

9 constraints on Esym (ρ0) and L from nuclear reactions and structures

Still within large uncertain region !!

The Nuclear Neutron Skin

For heavier stable nuclei: N>Z 1/22 1/22

n pr r

1/2 1/222 -np pnr rr Neutron Skin Thickness:

Bodmer, Nucl. Phys. 17, 388 (1960)

Sprung/Vallieres/Campi/Ko,

NPA253, 1 (1975)

Shlomo/Friedman, PRL39, 1180 (1977)

……

The Esym vs. Nuclear Neutron Skin

2 2

Neutron-Skin Thickness:

(fm)n pr rS 208( Pb) varies from 0.04 fm to 0.24 fm

depending on the Skyrme interaction !

S

Good linear Correlation: S-L

Chen/Ko/Li, PRC72,064309 (2005)

The Esym vs. Nuclear Neutron Skin

Chen/Ko/Li, PRC72,064309 (2005)

For heavier nuclei: Still good linear correlation between S-L

Neutron-Skin Thickness:

Pressure difference between n and p:

Slope of the Symmetry Energy:

n p

S

p p

L

( )n pS p p L

B.A. Brown, PRL85,5296 (2000)

The Skyrme HF Energy Density Functional

Standard Skyrme Interaction:

_________

9 Skyrme parameters:

9 macroscopic nuclear properties:

There are more than 120 sets of Skyrme- like Interactions in

the literature

Chen/Ko/Li/XuarXiv:1004.4672

Agrawal/Shlomo/Kim AuPRC72, 014310 (2005)

Yoshida/SagawaPRC73, 044320 (2006)

The Skyrme HF Energy Density Functional

Chen/Cai/Ko/Li/Shen/Xu, PRC80, 014322 (2009): Modified Skyrme-Like (MSL) Model

Chen/Ko/Li/Xu, arXiv:1004.4672

The Skyrme HF with MSL0

Chen/Ko/Li/Xu, arXiv:1004.4672

Correlations between Nuetron-Skin thickness and macroscopic Nuclear Properties

For heavy nuclei 208Pb and 120Sn: Δrnp is strongly correlated with L, moderately with Esym(ρ0), a little bit with m*s,0

For medium-heavy nucleus 48Ca:Δrnp correlation with Esym is much weaker; It further depends on GV and W0

Important Terms

Constraining Esym with Neutron Skin Data

Neutron skin constraints on L and Esym(ρ0) are insensitive tothe variations of other macroscopic quantities.

(~independent of Esym(ρ0))

A quite stringent constraint on Δrnp of 208Pb:

Constraining Esym with Neutron Skin Data and Heavy-Ion Reactions

N-Skin + HIC

Core-Crust transition density in

Neutron stars:

Global nucleon optical potential Esym(ρ0)=31.3 ± 4.5 MeV, L=52.7 ± 22.5 MeV

Xu/Li/Chen, arXiv:1006.4321v1, 2010

Esym: Global nucleon optical potential

Consistent with Sn neutron skin data!

Symmetry energy and Nuclear Effective Interaction

Chen/Ko/Li, PRC72,064309 (2005) Chen/Ko/Li, PRC76, 054316(2007)

L=58 ± 18 MeV: only 32/118 L=58 ± 18 MeV: only 8/23

We have proposed a novel method to explore transparently the correlation between observables of finite nuclei and nuclear matter properties.

The neutron skin thickness of heavy nuclei provides reliable information on the symmetry energy. The existing neutron skin data of Sn isotopes give important constraints on the symmetry energy and the neutron skin of 208Pb

Combining the constraints on Esym from neutron skin with that from isospin diffusion and double n/p ratios in HIC’s impose quite accurate constraint of L=58±18 MeV approximately independent of Esym

Our correlation analysis method can be generalized to other mean- field models (e.g., RMF) or density functional theories and a number of other correlation analyses are being performed (giant resonance, shell structure,,…… )

IV. Summary and Outlook

谢 谢！

(1) EOS of symmetric matter around the saturation density ρ0

GMR 0Frequency f K

Giant Monopole Resonance

K0=231±5 MeVPRL82, 691 (1999)Recent results:K0=240±10 MeVG. Colo et al. U. Garg et al.S. Shlomo et al.

__

0

22

0 0 2Incompressibility: K =9 ( )

d E

d

EOS of Symmetric Nuclear Matter

(2) EOS of symmetric matter for 1ρ0< ρ < 3ρ0 from K+ production in HIC’sJ. Aichelin and C.M. Ko,

PRL55, (1985) 2661C. Fuchs,

Prog. Part. Nucl. Phys. 56, (2006) 1

Transport calculations indicate that “results for the K+ excitation function in Au + Au over C + C reactions as measured by the KaoS Collaboration strongly support the scenariowith a soft EOS.”

C. Fuchs et al, PRL86, (2001) 1974

See also: C. Hartnack, H. Oeschler, and J. Aichelin,

PRL96, 012302 (2006)

EOS of Symmetric Nuclear Matter

(3) Present constraints on the EOS of symmetric nuclear matter for 2ρ0< ρ < 5ρ0 using flow data from BEVALAC, SIS/GSI and AGS

Use constrained mean fields to predict the EOS for symmetric matter

• Width of pressure domain reflects uncertainties in comparison and of

assumed momentum dependence.

P. Danielewicz, R. Lacey and W.G. Lynch, Science 298, 1592 (2002)

2Pressure P( )s

E

The highest pressure recorded under laboratory controlled conditions in nucleus-nucleus collisions

y

px

High density nuclear matter2 to 5ρ0

EOS of Symmetric Nuclear Matter

Solve the Boltzmann equation using test particle method Isospin-dependent initialization Isospin- (momentum-) dependent mean field potential

Isospin-dependent N-N cross sections a. Experimental free space N-N cross section σexp

b. In-medium N-N cross section from the Dirac-Brueckner approach based on Bonn A potential σin-medium

c. Mean-field consistent cross section due to m* Isospin-dependent Pauli Blocking

0 sym

1(1 )

2 z CV V V V

Phase-space distributions ( , , ) satify the Boltzmann equation

( , , ) ( , )p r r p c NN

f r p t

f r p tf f I f

t

Isospin-dependent BUU (IBUU) model

Transport model for HIC’s

EOS

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)

Transport model: IBUU04

Esym: Isospin Diffusion in HIC’s

How to measure Isospin Diffusion?

PRL84, 1120 (2000)

______________________________________

A+A,B+B,A+BX: isospin tracer

Isospin Diffusion/Transport

Isoscaling in HIC’sIsoscaling observed in many reactions

2 1

( ) /

Y / Yn pN Z Te

M.B. Tsang et al. PRL86, 5023 (2001)

Esym: Isoscaling in HIC’s

High density behaviors of Esym

n/p ratio of the high density region

Li/Yong/Zuo, PRC 71, 014608 (2005)Isospin fractionation!

Heavy-Ion Collisions at Higher Energies

Subthreshold K0/K+ yield may be a sensitive probe of the symmetry energy at high densities

Aichelin/Ko, PRL55, 2661 (1985): Subthreshold kaon yield is a sensitive probe of the EOS of nuclear matter at high densities

Theory: Ferini et al., PRL97, 202301 (2006) Exp.: Lopez et al. FOPI, PRC75, 011901(R) (2007)

96 9644 44

96 9640 40

Ru+ Ru and

Zr+ Zr@1.528 AGeV

K0/K+ yield is not so sensitive to the symmetry energy! Lower energy and more neutron-rich system???

High density behaviors of Esym: kaon ratio

IBUU04, Xiao/Li/Chen/Yong/Zhang, PRL102,062502(2009)

High density behaviors of Esym: pion ratio

A Quite Soft Esym at supra-saturation densities ???Zhang et al.,PRC80,034616(2009)

IDQMD, Feng/Jin, PLB683, 140(2010)

Pion Medium Effects?Xu/Ko/Oh

PRC81, 024910(2010)

Threshold effects?……

High density behaviors of Esym: n/p v2

A Stiff Esym at supra-saturation densities ???

W. Trauntmann et al., arXiv:1001.3867

Esym at very low densities: Clustering effects

S. Kowalski, et al., PRC 75 (2007) 014601.

Horowitz and Schwenk, Nucl. Phys. A 776 (2006) 55

Esym at very low densities: Clustering effects

J. B. Natowitz et al., arXiv:1001.1102

PRL, 2010