Determination of Density Dependence of Nuclear Matter Symmetry Energy in HIC’s

ISOSPIN PHYSICS AND LIQUID GAS PHASE TRANSITION, CCAST, Beijing, Aug. 19-21, 2005

Lie-Wen Chen(Department of Physics, Shanghai Jiao Tong

University)

Collaborators: V. Greco, C. M. Ko (Texas A&M University)

B. A. Li (Arkansas State University)

Nuclear Matter Symmetry Energy Two-Nucleon Correlation Functions Light Cluster Production and

Coalescence Model Isospin Transport/Diffusion Discussions Summary

Contents

References: PRL90, 162701 (2003); PRC68, 017601 (2003); PRC68, 014605 (2003); NPA729, 809(2003);PRC69, 054606 (2004); PRL94, 032701 (2005);Nucl-th/0508024.

Neutron Stars …

Structures of Radioactive Nuclei, SHE …

Isospin Effects in HIC’s …

Isospin in Intermediate Energy Nuclear Physics

Many-Body Theory

Many-Body Theory

Transport Theory General Relativity

Nuclear Force

EOS for Asymmetric

Nuclear Matter

Density Dependence of the Nuclear Symmetry Energy

HIC’s induced by neutron-

rich nuclei (CSR,GSI,

RIA,…)Pre-eq. n/p

Isospin fractionation

Isoscaling in MF

n-p differential transverse flowProton differential elliptic flow

π-/π+…

Isospin diffusionTwo-nucleon correlation functions

Light clusters (t/3He)

Thickness of neutron skin

Most uncertain property of an asymmetric

nuclear matter

Nuclear Matter Symmetry Energy

EOS of Asymmetric Nuclear Matter

s2 4

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

Isospin-Independent Part0 2 / 3

0

3( ,0) ( / )

2 1 5 F

a bE u u E u u

(Skyrme-like)

Nuclear Matter Symmetry Energy

0

sym 0

s

2

sym0 0sym 0

0 0

sym0

ym 0

( ) , ( )3 18

30 MeV (LD mass formula: )

( )3 (Many-Body Theory: : 50 200 M eV; Exp: ???)

( )

( ) Meyer & Swiatecki, NPA81; Pomorski & D

KE

E

L

K

ude

LE

E

k,

L

PRC67

0

2sym2

0 symsym

asy sym

2

isobaric incompres

( )9 (Many-Body Theory: : 700 466 MeV

The isospin part of the of asymmetric nuclear matter

(GMR : 566 1350 34 159M

siblity

( )6 eShlomo &Youngblood,PRC

EK

K L 47K

V)

Density dependence of the symmetry energy from SHF

2 2

L.W.Chen et (7 al40.4 ., (MeV) ( 2 )

(f

0.9)78. .

m

5 3 2

)n p

L S unpublis d

r r

he

S

BA Brown, PRL85

symThickness of neutron skin S vs. ( )E SkX~Variation Many-Body Theory

Most recent parameterization for studyingthe properties of neutron stars

sym sym 0( ) ( )E E u

H. Heiselberg& M. Hjorth-Jensen,Phys. Rep. 328(2000)

2 / 3 2sym sym

2 / 3 2sym 0

2 / 3sym 0

( , ) [ ( ( ) 12.7 ) ]

[ ( )( 1) 4.2 ]

2[ ( ) 12.7 ]

( (+) and ( ))

q

V E u

E u u

E u u

q n p

The symmetry potential acting on a nucleon

The neutron and proton symmetry potentials with the stiff (γ=2) and soft (γ =0.5) symmetry energies

γ =0.5:L=52.5 MeV and Ksym=-78.8 MeV γ=2.0: L=210.0 MeV and Ksym=630.0 MeV

Phenomenologically parameterizing the nuclear matter symmetry

energy

Isospin-dependent BUU (IBUU) model

Solve the Boltzmann equation using test particle method Isospin-dependent initialization Isospin-dependent mean field

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

Two-Nucleon Correlation Functions

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 rThe 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: including final-state nuclear and Coulomb interactions (Scott Pratt, NPA 566, 103 (1994))

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

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 sections

Chen,Greco,Ko,Li, PRL90, PRC68, (2003)

The covariant 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; NPA729

Butler,Pearson,Sato,Yazaki,Gyulassy,Frankel,Remler,Dove,Scheibl,Heinz,Mattiello,Nagle,Polleri,

Biro,Zimanyi,Levai,Csizmadia,Hwa,Yang,Ko,Lin,Voloshin,Molnar,Greco,Fries,Muller,Nonaka,Bass,…

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

0 1 2 3

21 1 2 1 3

10 0 0

22 1 2 2 3

20 0 0

23 1 3 2 3

30 0 0

The Lorentz Matrix

is the boosted four-ve

11 1 1

11 1 1

11

l

1 1

ocity .

b b b b

b b b b bb

b b b

b b b b bb

b b b

b b b b bb

b b b

b

L

Four-momentum: Four-coordina te : x x

y y

z z

E E t t

p p x x

p p y y

p p z z

L L

Dynamical coalescence model

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

Chen,Ko,Li, NPA729

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

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

Symmetry Energy Effects on t/3He ratio

50 100 150 2001.5

2.0

2.5

52Ca+48Ca, E=80 AMeV, b=0 fm =0.5 =2.0 =0.5 with soft EOS =0.5 with

medium

Y(t)/

Y(3 He

)

t (fm/c)

Stiffer symmetry energy gives smaller t/3He ratio With increasing kinetic energy, t/3He ratio increases for soft symmetry energy but slightly decreases for stiff symmetry energy

Isospin Transport/Diffusion

How to measure Isospin Transport?

PRL84, 1120 (2000)

______________________________________

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

( );

( )

0 S

1

trong

r

e

e

r

Weak

E=50 AMeV and b=6 fm

_____________IBUU04

MDI interaction

Chen,Ko,Li,

PRL94,2005

MDI ~Finite Range Gogny Interaction

Lane Potential Chen,Ko,Li, PRL93,2005

asy

GMR ( ):

: 566 1350 34 159MeV)

Shlomo &Youngblood,PRC47

K

1

2

3

4

5

6

7

8

P<300 MeV/c

(b) pp

P>500 MeV/c

(d) nn

(c) np

0.0

0.5

1.0

1.5

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

(a) nn

C(q

)

q (MeV/c)

(e) pp

10 20 30 40

1.0

1.5

2.0

2.5

3.0

3.5

4.0

MDI with soft sym. pot. MDI with hard sym. pot.

10 20 30 40 50

(f) np

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

Two-nucleon correlation functions

The sensitivity becomes weaker with momentum-dependence

1. Effects of momentum-dependence of nuclear potential

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

Discussions

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

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 stiffness of the symmetry energy

2. Effects of momentum-dependence of nuclear potential

Stiff Symmetry Energy:

MDI with 2

Soft Symmetry Energy:

MDI with 1x

x

3. Effects of in-medium cross sections on isospin transport

Li,Chen, Nucl-th/0508024.

np cross section is reduced in nuclear medium

3. Effects of in-medium cross sections on isospin transport

Ri(isospin transport/diffusion) Symmetry potential and np collisions

Li,Chen, Nucl-th/0508024.

asy

The parameter is found to be between 0.69 and 1.05

The K is norrowed down to 500 50 MeV, which agrees very well with

the giant resonance results about Sn isotopes (by Fujiwar

Compared wi

a)

208th the experimental data about the n-skin of Pb: 0.8

4. Have We Already Known the Density Dependence of Nuclear Matter Symmetry Energy at Sub-saturated

Densities?

W. D. Tian, Y. G. Ma, et al., Isoscaling + CQMD

__________________________________________________________________

arXiv:nucl-ex/0505011

Isocaling+AMD

sym

208

0

Isoscaling

E ( )=31.6( / )

0.7 is

Neutron-skin

Isospin Transport/Diffusion:

of Pb: 0.8

+AMD: 0.6 1

most ac

0.69 1

ce

.

.05

pta

05

ble

5. The High Density Behaviors of Nuclear Matter Symmetry

Li,Chen,Ko,Yong,Zuo, nucl-th/0504008; Li,Chen,Das, Das Gupta,Gale,Ko,Yong,Zuo, nucl-th/0504069

B. A. Li, PRL88 (2002) 192701

nucl-th/0504065, Phys.Rev. C71 (2005) 054907

Other possible observations: Kaons, Σ, …

———————————————————————————————————————————————————————————

6. Momentum Dependence of Symmetry Potential

Recent progress:E.N.E. van Dalen, C. Fuchs, A. Faessler, NPA744, (2004); PRL95,(2005)Zhong-yu Ma, Jian Rong, Bao-Qiu Chen, Zhi-Yuan Zhu, Hong-Qiu Song, PLB604, (2004)F. Sammarruca, W. Barredo, P. Krastev, PRC71, (2005)W. Zuo, L.G. Cao, B. A. Li, U. Lombardo, C.W. Shen, PRC72, (2005)L.W. Chen, C.M. Ko, B.A. Li, to be submitted

‘Puzzle’?

Di Toro et al.

Summary

Two-particle correlation functions and t/3He ratio are useful probes of the nuclear symmetry energy

The sub-saturated density behavior of the symmetry energy become more and more clear from the isospin diffusion and isoscaling, and n-skin of Pb

The high density behavior of the symmetry energy and the momentum dependence of the symmetry potential need much further effort

Thank you!谢谢大家！