重离子碰撞与核物质状态方程

陈列文( 上海交通大学物理系 lwchen@sjtu.edu.cn)

2009 威海高能物理暑期论坛暨“ RHIC 物理和 CSR 强子物理”研讨会， 2009 年 8 月 8 日 -14 日，威海，山东

目录

非对称核物质的状态方程与对称能同位旋相关的重离子碰撞微观输运模型重离子碰撞：对称核物质的状态方程重离子碰撞：对称能的低密行为重离子碰撞：对称能的高密行为对称能对其他物理量的约束总结和展望

Main References: L.W. Chen, C.M. Ko, B.A. Li, and G.C. Yong, Front. Phys. China 2(3), 327 (2007)B.A. Li, L.W. Chen, and C.M. Ko, Phys. Rep. 464, 113-281 (2008)

一、非对称核物质的状态方程与对称能极端条件下的核物理

Energy,Temperature,Density:Phase Transitions:Liquid-Gas;Chiral RestorationQGP-Hadrons……

Isospin:Nuclei far from Beta-Stability Line……

Spin:High Spin State and,SD (Super-Deformed), HD (Hyper-Deformed)……

A and Z:SHE(Super-HeavyElements)……

HIC’s

同位旋核物理 : 核结构

18 18

12

12

12

3

( , )n pE

Isospin asymmetry δ=(ρn-ρp)/ρ

ρ=ρn+ρp

densitySymmetric

matte

r

ρ n=ρ p

Studied extensivelyStudied extensively

A new dimension

A new dimension

Very small isospin asymmetry ???

同位旋核物理 : 中能重离子碰撞

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

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

tions???

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!!!

同位旋核物理 : 对称能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).

Isospin physicsIsospin physics

重离子加速器装置1. HIRFL, CSR/HIRFL (China)2. GANIL (France)3. GSI (Germany)4. NSCL/MSU,FRIB/MSU5. RIKEN (Japan)

1 50 100 150 200 250 300

10

102

103

104

A

E (

MeV

/u)

HIRFL

GANIL

NSCL/MSU

CSR/HIRFL

RIKENGSI

Dubna, LBL, ORNL, TAMU,INFN, KVI,…

AGS,RHIC/BNLSPS,LHC/CERN

中能重离子加速器装置

Radioactive beam facilities are being built around the world

IMP CIAE

Providing new opportunities for both nuclear physics and astrophysics

World status of Rare Isotope Accelerators

核物质的状态方程

-

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 P

incompessibilt

of the nuclear matter can be expressed as

The of the nuclear matter can be expressed ay K

s

T N

P T

, constant

30Saturation d of symmetric nuclear matter at T=0 MeV:

of symmetric nuclear mat

Empirical values abo

ut the nuclear matte

( , , ) 9

0.16 fmensity

Pressu

r EO

terr

S

e at

:T N

PK T

3

0 0

0

0

0

T=0 MeV:

The of symmetric nuclear matter at and T=0 Menergy of per nucleon

Incompes

eV:

of symmetric nucl

( ) 0 MeVfm

16 MeV/nu

ear matter at T=0

cle

MeV

on

200 400 : si Mbilty eV

P

K

Liquid-drop model

核的对称能

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)

核物质的对称能EOS of Asymmetric Nuclear Matter

s2 4

ym ( )( , ) ( ), ( ),0) /( n pE OE E (Parabolic law)

The Nuclear Symmetry Energy2

sym 2

1 ( , )( )

2

EE

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

2sym2

0 sym2

asy sym

sym

(Sharma et al

isobaric incompressiblity

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

The isospin part of the of asymmetric nuclear matter

700 466 MeV

320 180 6 GMR( : M eV

K

K K K

K

L

E

Shlomo&Youngblood,PRC47,529(93);

., PRC38, 2562 (88));

566 1350 34 159 M eV

550 1

(

(T. Li et al, PRL99,162503(2007))00 MeV )

Symmetry energy termSymmetric Nuclear Matter

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 ……

Nuclear Matter EOS: Many-Body Approaches

Saturation Pionts of Symmetric Nuclear Matter from Microscopic Many-Body Approaches: The Coester Line

Nuclear Matter EOS: Isospin Dependence

I. Tanihata Preprint RIKEN-AF-NP-229, July, 1996.N/Z=0,0.2,0.4,0.6,0.8,1(From top to bottom)

Z.H. Li, (2007)

Van Dalen/Fuchs/Faessler EPJA31, 29 (2007)

SIII TM1

核物质的对称能

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

二、同位旋相关的重离子碰撞微观输运模型

Broad applications of transport modelsin astrophyics, plasma physics, electron transport in semiconductor and nanostructures, particle and nuclear physics, nuclear stockpile stewardship

Transport Models for HIC’s at intermediate energies:

N-body approachesCMD, QMD,IQMD,IDQMD,ImQMD,ImIQMD,AMD,FMD

One-body approachesBUU, BNV, LV, IBL RBUU,RVUU,…

Transport Models

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

2*

/ NNmedium free

NN

in neutron-rich matter/medium free

*NN is the reduced mass of the

colliding pair NN in mediumJ.W. Negele and K. Yazaki, PRL 47, 71 (1981)V.R. Pandharipande and S.C. Pieper, PRC 45, 791 (1992)M. Kohno et al., PRC 57, 3495 (1998)D. Persram and C. Gale, PRC65, 064611 (2002).

1. In-medium cross sections are reduced2. nn and pp cross sections are splitted due to the neutron-proton effective mass sli

tting in neutron-rich matter

Li/Chen, PRC72 (2005)064611

Medium effects: effective mass on the incoming current in initial state and level density of the final state

Neglecting medium effects on the transition matrix

In-medium Nucleon-nucleon cross sections:Effective mass scaling model

三、重离子碰撞：对称核物质的状态方程Why Heavy-Ion Collisions?

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!

LW Chen et al., PRC80, 014322 (2009)

(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.

__

0

22

0 0 2Incompressibility: K =9 ( )

d E

d

对称核物质的状态方程

对称核物质的状态方程(2) EOS of symmetric matter for 1ρ0< ρ < 3ρ0 from K+ production in HIC’s

J. Aichelin and C.M. Ko, PRL55, (1985) 2661

C. 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)

(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

对称核物质的状态方程

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) in Nuclear Reactions (an incomplete list !)

四、重离子碰撞：对称能的低密行为

对称能探针：同位旋弥散

How to measure Isospin Diffusion?

PRL84, 1120 (2000)

______________________________________

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

Isospin Diffusion/Transport

对称能探针：同位旋弥散

int

Isospin diffusion in 124Sn+112Sn@E/A=50 MeV and b=6 fm

The theoretical analysis (BUU) did NOT include Momentum Dependence for nuclear potential !

0A BXR

( );

( )

0 S

1

trong

r

e

e

r

Weak

If complete isospin mixing/ If complete isospin mixing/ equilibriumequilibrium

对称能探针：同位旋弥散

Chen/Ko/LiPRL94, (2005) 032701

0 50 100 150

0.0

0.5

1.0

1.5

/0

x=-1 MDI SBKD /

0 (MDI)

/0 (SBKD)

t (fm/c)

124Sn + 112Sn

MSU data

Ri

Momentum-independent Momentum-dependent

Comparing momentum-dependent IBUU04 clculations with data on isospin transport

All have the same Esym (ρ)=31.6 (ρ/ρ0)1.05

对称能探针：同位旋弥散

Li/ Chen, PRC72, 064611(2005)

Symmetry energy, isospin diffusion, in-medium cross section

Isospin Diffusion Data Kasy=-500±50 MeVL=86±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)

对称能探针：同位旋标度Isoscaling in HIC’s

Isoscaling observed in many reactions

2 1

( ) /

Y / Yn pN Z Te

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

对称能探针：同位旋标度

Consistent with isospin diffusion data!

Constraining Symmetry Energy by Isocaling: TAMU Data

Shetty et al.PRC75(07);PRC76(07)

IBUU04 : S~31.6(/o)

ImQMD: n/p ratios and two isospin diffusion measurements

Consistent constraints from the 2 analysis of three observables

S=12.5(/o)2/3 + 17.6 (/o)

i

i

i

i i

对称能探针：同位旋弥散和双 n/p 比率

Tsang/Zhang/Danielewicz/Famiano/Li/Lynch/Steiner, PRL 102, 122701 (2009)

1.05

00

0.69 31.6( /31.6( / ) )( )

Symmetry energy constrained at -saturation densities

between the and lines, agrees extremely well with the

sub

-1 0 PR

AsymE

x x

(IBUU04)

(ImQMD)

X=-1

Tsang et al., PRL 102, 122701 (2009)

对称能的亚饱和密度行为

Chen/Ko/Li, PRL 94, 032701 (2005)

( 亚饱和密度： 0.2-0.3<ρ/ρ0<1.2)

极端低密时的对称能：结团效应

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

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

对称能探针：轻粒子产生

The coalescence model

3

1 131

The element of a spacelike hypersurface

1 2

( ;

at freeze-out

Coalescence pr

) ( , , ; , , )(2 )

obability (Wigner phase-space

:

densi

: ty)

MWi

C C i i i i i C M Mi i

i

WC

M C

d pN g p d f x p x x p p

E

d

Chen/Ko/Li, PRC68, 014605 (2003); NPA729, 809 (2003)

Depends on constituents’ space-time structure at freeze-out Neglecting the binding energy effect (T>>Ebinding),

Coalescence probability: Wigner phase-space density in the rest-frame of the cluster.

Rare process has been assumed (the coalescence process can be treated perturbatively).

Higher energy collisions and higher energy cluster production!

Light Cluster Production and Coalescence Model

对称能探针：轻粒子产生

10-5

10-4

10-3

10-2

10-1

data (b=6-7 fm) IBUU+Coalescence

(b=6.5 fm)

Ekin

(MeV)

(a) Deuteron

(d) Deuteron

10-7

10-6

10-5

10-4

10-3

10-2

(b) Triton

36Ar+58Ni@E/A=95 MeV, 60o<c.m.

<120o

data (b=4-5 fm) IBUU+Coalescence

(b=4.5 fm)

dM

/dE

kin (

MeV

-1)

(e) Triton

0 50 100 150 200 25010-7

10-6

10-5

10-4

10-3

10-2

(c) 3He

0 50 100 150 200 250 300

(f) 3He

Isospin symmetric collisions at E/A≈100 MeV

Deuteron energy spectra reproduced Low energy tritons slightly underestimated Inverse slope parameter of 3He underestimated; probably due to neglect of

• larger binding effect• stronger Coulomb effect• wave function

Data are taken from INDRA Collaboration (P. Pawlowski, EPJA9)

Try Coalescence modelat intermediate energies!

Chen/Ko/Li, NPA729, 809 (2003)

对称能探针：轻粒子产生

Symmetry Energy Effects on t/3He ratio

Stiffer symmetry energy gives smaller t/3He ratio

Stiff: MDI with 2

Soft: MDI with 1

x

x

对称能探针：核子 - 核子关联函数

The two-particle correlation function is obtained by convoluting the emission function g(p,x), i.e., the probability of emitting a particle with momentum p from space-time point x=(r,t), with the relative wave function of the two particle, i.e.,

24 41 2 1 2

4 41 1 2 2

1 2 1 2

( / 2, ) ( / 2, ) ( , )( , )

( / 2, ) ( / 2, )

, ( ) / 2

( , ) is the relative two-particle wavefunction

d x d x g x g xC

d x g x d x g x

P P q rP q

P P

P p p q p p

q r

The two-particle correlation function is a sensitive probe to the space-time structure of particle emission source by final state interaction and quantum statistical effects (φ(q,r))

Correlation After Burner (Crab): including final-state nuclear and Coulomb interactions (S. Pratt, NPA 566, 103 (1994))

How to detect the space-time structure of nucleon emission experimentally?

Two-Nucleon Correlation Functions

对称能探针：核子 - 核子关联函数

High momentum nucleons emitted earlier than low momentum ones Earlier emissions for stiffer symmetry energy Larger separation in neutron and proton emission times for softer symmetry energy

Space-time structure of nucleon emissionEmission times from IBUU

Chen/Ko/Li, PRL90, 162701 (2003)

对称能探针：核子 - 核子关联函数

MDIDas, Das Gupta, Gale and LiPRC67, (2003)

Effects of momentum-dependence of nuclear potential

Pairs with P>500 MeV:n-p CF: 11%

Stiff Symmetry Energy: MDI with 2

Soft Symmetry Energy: MDI with 1x

x

The isospin effects on two-particle correlation functions are really observed in recent experimental data !!!R. Ghetti et al., PRC69 (2004) 031605肖志刚等 , PLB, (2006)

Chen/Ko/Li, PRC69, 054606 (2004)

五、重离子碰撞：对称能的高密行为

n/p ratio of the high density region

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

Heavy-Ion Collisions at Higher Energies

对称能高密探针 :pion 比率

0

nn 0 1 5 a) Δ(1232) resonance model pp 5 1 0 in first chance NN scatterings: np(pn) 1 4 1 (negelect rescattering and reabsorption)

2

2

2

)(5

5ZN

NZZ

NZN

R. Stock, Phys. Rep. 135 (1986) 259. b) Thermal model: (G.F. Bertsch, Nature 283 (1980) 281; A. Bonasera and G.F. Bertsch, PLB195 (1987) 521)

exp[( ) / ]n p kT

H.R. Jaqaman, A.Z. Mekjian and L. Zamick, PRC (1983) 2782.

c) Transport models (more realistic approach): see, e.g., Bao-An Li, Phys. Rev. Lett. 88 (2002) 192701.

31 1( ) {ln ( ) ( )}

2

m mn p mnn p asy asy Coul m T n p

mp

mV V V kT b

m

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

对称能高密探针 :pion 比率

A Quite Soft Esym at supra-saturation densities !!!M. Zhang et al., arXiv:0904.0447

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: Famiano et al., PRL97, 052701 (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???

对称能高密探针 :Kaon 比率

对称能高密探针：挤出 n/p 比率In the squeeze-out direction: nucleons emitted from the high density participant region have a better chance to escape without being hindered by the spectators. These nucleons thus carry more direct information about the high density phase of the reaction.

Yong/Li/Chen, nucl-th/0703042PLB650, 344 (2007) The effect can be 40% at higher pT !

对称能高密探针 : n-p 差分流

Neutron-proton differential flow

( )

1

1( )

( )

i N yx x

n p i ii

F y pN y

y

px

px

yy

symmetry potential is generallyrepulsive for neutrons and attractive for protons

1i for n and p

Bao-An Li, PRL 85, 4221 (2000).

Yong/Li/Zuo, High energy physics and nuclear physics (2005).

Isospin asymmetry of free nucleons

六、对称能对其他物理量的约束

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

(1) Symmetry Energy from SHF and RMFChen/Ko/Li, PRC76, 054316(2007)

Only 4/21! Only 5/23!

对称能：中子皮(2) Neutron-Skin of 208Pb from SHF

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)

208

132

124

90

86 25 MeV :

0.21

0.

( ) : 0.22 0.04 fm ( );

( ) : 0.29 0.04 fm ( );

( ) : 0

Pb

Sn

Sn

PRL95, 2005

.22 0.04 fm ( )

( ) : 0.088 0.04 fm

27

0.19

0. ( )

(

09

RM ,

Neut

Todd

ron-skin

-Rutel/PF i

thickness

ekarewic

)

:

z,

S

S

S

L

ZrS

Lattimer/Prakash, Science 304, 536 (2004)

Neutron star has solid crust over liquid core.Rotational glitches: small changes in period from sudden unpinning of superfluid vortices. Evidence for solid crust. 1.4% of Vela moment of in

ertia glitches. Needs to know the transitio

n density to calculate the fractional moment of inertia of the crust

Link et al., PRL83,3362 (99)

core-crust transition

对称能：中子星的壳芯转变密度

Kazuhiro Oyamatsu, Kei Iida

Phys. Rev. C75 (2007) 015801

pasta

Xu/Chen/Li/Ma, PRC79, 035802 (2009)

Xu/Chen/Li/Ma, ApJ 697, 1547 (2009), arXiv:0901.2309

Parabolic Approximation has been assumed !!!

Significantly less than their fiducial values:ρ t=0.07-0.08 fm-3 and Pt=0.65 MeV/fm3

对称能：中子星的壳芯转变密度

The softest symmetry energythat the TOV is still stable is x=0.93 giving M_max=0.11 solar mass and R=>28 km

For pure nucleonic matter?Hyperon? Quark?New Physics???

Soft symmetry energy at HD ?K

0=211 MeV is used, higher incompressibility

for symmetric matter will lead to higher masses systematically

?

对称能的高密行为：中子星？

Medium effects on pion production???Xu/Ko/Oh, arXiv:0906.1602

七、总结和展望

The isospin diffusion data, Isoscaling and double n/p ratio seem to give a stringent constraint for the sub-normal density behavior of the symmetry energy (L=86±25 MeV and Kasy=-500±50 MeV) (Crosscheck is definitely needed !!!) Pion ratio data from FOPI favors a supersoft symmetry energy at super-saturation densities. Probing the high density behavior of the symmetry energy remains a big challenge and CSR/Lanzhou can make important contribution. Experiments: detectors (n,p,fragments,pi,kaons,photons… ) The transport model provides an important tool to study equation of state of asymmetric nuclear matter, especially for the nuclear symmetry energy: How to treat fragments? Clustering effects? Medium effects? Many-body collisions? Lorentz Covariance? Self-consistency?……

谢 谢！

Development and Challenges of Transport Models

1. Development of transport models Transport codes often implement extra physical assumptions and dynamical mechanisms which go beyond the equations used to motivate their designs. These algorithms often undergo evolutions with time as we make progresses in our R&D efforts and also as our needs for including new processes arise. They may involve many phenomenological parameters which are not all well experimentally constrained yet because of the lack of the relevant experimental data, and some of them are exactly what we want to learn.2. Challenges of transport models for reactions involving radioactive beams Develop practically implementable quantum transport theories ( the de Broglie wavelength may be comparable to the nucleon mean free path in energetic central reactions, and the uncertainty principle imposes a strong correlation between delocalization of nucleons and their momentum distribution) Include more structure information in the initial state especially for peripheral reactions. Use consistently all inputs (initial state, mean field and бNN) from the same interactions

3. Self-consistent Isospin dependent Lorentz Covariant transport models for HIC’s at higher energies

4. Many-body collisions and clustering effects

放射性核束装置

Cooling Storage Ring (CSR) Facility at HIRFL in China up to 500 MeV/u for 238U http://www.impcas.ac.cn/zhuye/en/htm/247.htm.

Radioactive Ion Beam (RIB) Factory at RIKEN in Japan http://www.riken.jp/engn/index.html

FAIR/GSI in Germany up to 2 GeV/u for 132Sn http://www.gsi.de/fair/index_e.html.

SPIRAL2/GANIL in France http://ganinfo.in2p3.fr/research/developments /spiral2

Facility for Rare Isotope Beams (FRIB) in USA up to 200 MeV/u for 132Sn http://www.frib.msu.edu

对称能探针：轻粒子产生

3( , ) Re ( / 2) ( / 2)W ik rd r k d r R r R

Hulthen wave function

23/ 415

21

( ) 2( )

2 ( )i

r rri

ii

e er c e

r

1

1

2

0.23 fm

1.61 fm

1.89 fmr

Wigner phase-space density for Deuteron

0.0 0.5 1.0 1.5 2.0 2.50

1

2

3

4

(k)

k (1/fm)

0 2 4 6 8 10 120.0

0.1

0.2

0.3

0.4 Hulthen Hulthen wih 15 Gaussians

(r)

r (fm)

Wigner transformation

对称能探针：轻粒子产生

3

3

2 2 2 2 2 2 2 2 21 2 1 2t/ He

2 2 22 1 2 3 2 3 1 3 1 2

t/ He1 2 3 1 2 3

1 21 1 3

1 2 1

3

2

( , ; , ) 8 exp( / / )

1 ( ) ( ) ((t:

)

2 ( )

1 3( ), ( ) (Jacobi Trans

format22

1.61 fm; He: 1.74 fm)

W k k

m m m m m m m m mr

m m m m m m

m m

m m m m

2 2

ρ λ k k

ρ r r λ r r r

2 1 3 3 1 21 2 1 2 3

2 1 2 11 1 2 2

11

1 21 2 1 2 3

ion)

2 6( ), ( ( ) )

2( )

( ) and ( ) with

1 1 3 1 12 and

2

m m m m m mm m m m m

m m m m m

1 2 1 2 2k k k k k k k

t/3He Wigner phase-space density and root-mean-square radius:

Wigner phase-space density for t/3He

Assume nucleon wave function in t/3He can be described by the harmonic oscillator wave function, i.e.,

3/ 421

( ) exp( )2 2

with the harmonic oscillator frequency

mm r

r

对称能探针：轻粒子产生

0 20 40 60 80 1001.5

2.0

2.5

52Ca+48Ca E=80 AMeV, b=0 fm

Soft Sym. Pot. Hard Sym. Pot.

(a) SBKD

Y(t

)/Y

(3 He

)

Ek (MeV)

0 20 40 60 80 100 120

(b) MDI

t/3He ratio

Still sensitive to the symmetry energy

Effects of momentum-dependence of nuclear potential

Stiff Symmetry Energy:

MDI with 2

Soft Symmetry Energy:

MDI with 1x

x

Chen/Ko/Li, PRC69, 054606 (2004)

对称能探针：核子 - 核子关联函数

Pairs with P>500 MeV:n-n CF: 20%p-p CF: 20%n-p CF: 30%

Symmetry Energy Effects on Two-Nucleon Correlation Functions

Effects are very small for both isoscalar potential and N-N cross sectionsChen/Ko/Li,

PRL90, 162701 (2003)

对称能探针：同位旋标度

sym is the symmetry energy of the finite nuclei or

bulk nuclear matter symmetr (Debaty ene ing trg oy

C

? pic)

Li/Chen, PRC74,034610 (2006) Xu/Chen/Li/Ma, PRC75,014607 (2007)

四、对称能对其他物理量的约束

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

Nuclear Matter Symmetry Energy from SHF

In 21 Skyrme interaction

parameter sets, only 4

parameter sets are consistent

with isospin diffusion da

Skyrme In

SIV,

terac

SV,

ta

G

:

tio

and R

ns:

四、对称能对其他物理量的约束

Non-linear RMF(10):NL1,NL2,NL3,NL-SH,TM1,PK1,FSU-Gold.

HA,NLρ,NLρ

Density-dependent RMF(7):TW99,DD-ME1,DD-ME2

PKDD,DD,DD-FDDRH-corr

Point-Coupling RMF(6):PC-F1,PC-F2,PC-F3,

PC-F4PC-LAFKVW

asy

TM1, NL , NL , PKDD, PC-LA, FKVW (6)

NL3, NL-SH, TM1,PK1, HA, NL , NL , TW99, PKDD,

D

88 25 MeV :

K 500 50 MeV (or 5

D-F, PC-F1, PC-F2,PC-F3, PC-F4, and

50 100) :

8

FKVW(15)

8 25 M

L

L

asyeV + K 500 50 (or 550 60) MeV

TM1, NL , NL , PKDD, FKVW (

5)

:

Chen/Ko/Li, PRC76, 054316(2007)

Nuclear Matter Symmetry Energy from RMF

四、对称能对其他物理量的约束Neutron-Skin of 208Pb from SHF

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)

四、对称能对其他物理量的约束

208

132 124

90

Neutron-skin thickness

( ) : 0.22 0.04 fm ( );

( ) : 0.2

86 25 MeV :

0.21

0.29 0.04 fm ( ); ( ) : 0.22 0.04 fm

Pb

Sn S7 0. ( )

( ) : 0

1

.0

n

:

9

S

S S

S

L

Zr

88 0.04 fm ( )

( , B.G. Todd-Rutel and J. Piekarewicz PRL

0.09

R 95, 2, )MF 005

Chen/Ko/LiPRC72,064309 (05)

208

x-ray cascade from antiprotoni

Pb

Klc atoms, os et al., PRC 76 (2000.16 0

( ) : ( )

fm 7 . (0 )6

S

Recent data on neutron - skin thickness :

0.16 fm→ L=40 MeV; 0.18 fm→ L=54.8 MeV

132

);

fm ( );

( )

014311

Klimkiewicz et al. PRC 76 (2007) 051603 (R)

Sn

0.18 0.035

0.24 0.04

: ( )

fm (

pygmy dipole resonances,

pygmy dipole resonances, Klimkiewicz et al. PRC 76 (2007) 05160

S

0.24 fm→ L=49.7 MeV

124

90

3 (R)

Sn

Terashima et al., PRC 77 (2008) 02

);

0.185 0.0

( ) : ( )

fm ( )

( ) :

17

0.07 0. fm 04

proton elastic scattering,

spin-dipole sum rul

4317

Yako/Sagawa/Sake, ai, PRC 74 (2006) 0( 5

S

S Zr

0.185 fm→ L=53.2 MeV

1303 (R))

Constrain N-Skin from obtained Esym

Lattimer/Prakash, Science 304, 536 (2004)

Neutron star has solid crust over liquid core.Rotational glitches: small changes in period from sudden unpinning of superfluid vortices. Evidence for solid crust. 1.4% of Vela moment of in

ertia glitches. Needs to know the transitio

n density to calculate the fractional moment of inertia of the crust

Link et al., PRL83,3362 (99)

core-crust transition

四、对称能对其他物理量的约束

Thermodynamic approach

Or , similarly one can use the RPA

If one uses the parabolic approximation (PA)

Then the stability condition is:

>0

均匀电中性 beta 稳定物质的不稳定性

Dynamical approachk0 (neglecting Coul.)

Stability condition:

壳芯 (core-crust) 转变密度：抛物线近似失效 !

(1) It is NOT enough to know the symmetry energy, one almost has to know the exact EOS of n-rich matter

Why?Because it is the determinant of the curvature matrixthat determines the stability condition Example:

Not so surprise:

Zhang/Chen, CPL 18 (2000) 142Steiner, Phys.Rev. C74 (2006) 045808

Higher-order term effects on direct URCA

Xu/Chen/Li/Ma, PRC79, 035802 (2009)

(2) Locating the inner edge of neutron star crust

Kazuhiro Oyamatsu, Kei Iida

Phys. Rev. C75 (2007) 015801

pasta

Xu/Chen/Li/Ma, PRC79, 035802 (2009)

Xu/Chen/Li/Ma, ApJ, in press (2009), arXiv:0901.2309

Parabolic Approximation has been assumed !!!

Significantly less than their fiducial values:ρ t=0.07-0.08 fm-3 and Pt=0.065 MeV/fm3

(3) Constraints on M-R relation of NS

(Isospin Diff)

(Empirical estimate Link et al., PRL83,3362(99))

(4) Properties of neutron star crusts

Xu/Chen/Li/Ma, ApJ, in press (2009), arXiv:0901.2309

Larger L leads to thicker neutron-skin , but thinner neutron star crust !!!

The softest symmetry energythat the TOV is still stable is x=0.93 giving M_max=0.11 solar mass and R=>28 km

For pure nucleonic matter???Soft symmetry energy at HD ?

K0=211 MeV is used, higher incompressibility

for symmetric matter will lead to higher masses systematically

?

(5) HD Esym and properties of neutron stars