Transcript Probing Density Dependence of Symmetry Energy in N

ISOSPIN PHYSICS AND LIQUID GAS PHASE
TRANSITION, CCAST, Beijing, Aug. 19-21, 2005
Determination of Density Dependence of
Nuclear Matter Symmetry Energy in HIC’s
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)
Contents
 Nuclear Matter Symmetry Energy
 Two-Nucleon Correlation Functions
 Light Cluster Production and
Coalescence Model
 Isospin Transport/Diffusion
 Discussions
 Summary
References:
PRL90, 162701 (2003);
PRC68, 014605 (2003);
PRC69, 054606 (2004);
Nucl-th/0508024.
PRC68, 017601 (2003);
NPA729, 809(2003);
PRL94, 032701 (2005);
Isospin in Intermediate Energy Nuclear Physics
Transport Theory
Isospin Effects in HIC’s …
EOS for
Asymmetric
Nuclear Matter
General Relativity
Neutron Stars …
Many-Body Theory
Most uncertain
property of an
asymmetric
nuclear matter
Nuclear Force
Many-Body Theory
Structures of Radioactive Nuclei, SHE …
Pre-eq. n/p
Isospin fractionation
Isoscaling in MF
π-/π+…
Density Dependence of
the Nuclear Symmetry Energy
Thickness of neutron skin
n-p differential transverse flow
HIC’s
induced by
neutronrich nuclei
(CSR,GSI,
RIA,…)
Light clusters (t/3He)
Isospin diffusion
Two-nucleon correlation functions
Proton differential elliptic flow
Nuclear Matter Symmetry Energy
EOS of Asymmetric Nuclear Matter
E(,  )  E(,0)  Esym ( ) 2  O( 4 ),   (n   p ) /  (Parabolic law)
Isospin-Independent Part
E (  ,0) 
a
b  3 0 2/3
u
u  EF u
2
1
5
(u   /  0 )
Nuclear Matter Symmetry Energy
(Skyrme-like)
2
L    0  Ksym    0 
Esym (  )  Esym ( 0 )  


 , (  0 )
3  0  18  0 
Esym ( 0 )  30 MeV (LD mass formula: Meyer & Swiatecki, NPA81; Pomorski & Dudek, PRC67 )
L  3 0
Esym (  )
Ksym  9 

2
0
(Many-Body Theory: L : 50
200 M eV; Exp: ???)
  0
 2 Esym (  )

(Many-Body Theory: Ksym : 700
2
466 MeV
  0
The isospin part of the isobaric incompressiblity of asymmetric nuclear matter
K asy  Ksym  6 L (GMR (Shlomo &Youngblood,PRC47 ):  566  1350 34  159MeV)
Density dependence of the symmetry energy from SHF
50
45
0.30
(a)
Skyrme-Hartree-Fock
with 21 parameter sets
(b)
208
0.25
40
(c)
Pb
SkX
0.20
30
S (fm)
Esym() (MeV)
35
25
20
0.15
0.10
15
10
0.05
5
0
0.00
0.00
0.05
0.10
0.15
0.20
0.25
0.30
-3
 (fm )
-60 -30 0
30 60 90
L (MeV)
SkX~Variation Many-Body Theory
-600 -400 -200
Ksym (MeV)
0
26
28
30
rn2 
rp2
(fm)
34
36
Esym(0) (MeV)
Thickness of neutron skin S vs. Esym (  )
BA Brown, PRL85
L  78.5  3.2  (740.4  20.9) S (MeV) (L.W.Chen et al., unpublished )
S
32
Phenomenologically parameterizing
the nuclear matter symmetry energy
H. Heiselberg&
M. Hjorth-Jensen,
Phys. Rep. 328(2000)
Most recent parameterization for studying
60
the properties of neutron stars
(a) =0.2
Esym (  )  Esym ( 0 )  u
Vsym (  ,  ) 

[  ( Esym (  )  12.7u 2 / 3 ) 2 ]
 q
 [ Esym ( 0 )(  1)u  4.2u 2 / 3 ] 2
 2[ Esym ( 0 )u  12.7u 2 / 3 ]
( q  n (+) and p ( ))
=0.5
=2
40
neutrons
20
Vsym (MeV)
The symmetry potential acting on
a nucleon
(b) =0.4
neutrons
0
protons
-20
protons
-40
-60
0.0
0.5
1.0
1.5
0.0
0.5
1.0
/0
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
1.5
2.0
Isospin-dependent BUU (IBUU) model
Phase-space distributions f ( r , p, t ) satify the Boltzmann equation
f ( r , p, t )
  p   r f   r   p f  I c ( f , NN )
t
 Solve the Boltzmann equation using test particle method
 Isospin-dependent initialization
 Isospin-dependent mean field
1
V  V0  (1   z )VC  Vsym
2
 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
Two-Nucleon Correlation Functions
How to detect the space-time structure of
nucleon emission experimentally?
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.,
d

C (P, q) 
4
x1d x2 g (P / 2, x1 ) g (P / 2, x2 )  (q, r)
4
2
4
4
d
x
g
(
P
/
2,
x
)
d
x2 g (P / 2, x2 )
1 
 1
P  p1  p2 , q  (p1  p2 ) / 2
 (q, r ) is the relative two-particle wavefunction
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: including final-state nuclear and Coulomb
interactions (Scott Pratt, NPA 566, 103 (1994))
Symmetry Energy Effects on Two-Nucleon Correlation Functions
8
7
(a) nn
P<300 MeV/c
7
6
52
5
(d) nn
P>500 MeV/c
5
48
Ca+ Ca
E=80 AMeV, b=0 fm
4
52
(d) nn
P>500 MeV/c
48
Ca+ Ca
E=80 AMeV, b=0 fm
4
3
3
2
2
1
1.5
1
(b) pp
(e) pp
C(q)
1.5
C(q)
(a) nn
P<300 MeV/c
6
1.0
(b) pp
(e) pp
(c) np
(f) np
1.0
0.5
0.5
0.0
0.0
4
=0.5
=2
3
(c) np
(f) np
3
=0.5 with K0=380 MeV
=0.5 with K0=210 MeV
=0.5 with K0=380 MeV
and in-medium
2
2
1
1
10
20
30
40
50
10
q (MeV/c)
20
30
40
50
0
10
20
30
40
0
10
20
30
q (MeV/c)
Pairs with P>500 MeV:
Effects are very small for
n-n CF: 20%
both isoscalar potential and
p-p CF: 20%
N-N cross sections
n-p CF: 30%
Chen,Greco,Ko,Li, PRL90, PRC68, (2003)
40
50
Light Cluster Production and Coalescence Model
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
,…
The covariant coalescence model
1  2    M  C
d 3 pi
W
N C  gC   pi  d i
f
(
x
;
p
)

( x1 , , xM ; p1 , , pM )
i
i
i
C
3
(2 ) Ei
i 1
M
d i : The element of a spacelike hypersurface at freeze-out
CW : Coalescence probability (Wigner phase-space density)
 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!
Chen,Ko,Li, PRC68; NPA729
Dynamical coalescence model
The Lorentz Matrix
b1
b2
b3 
 b0


2
b
b
b
b
b
1 2
1 3
 b 1  1

1

1  b0
1  b0
1  b0 


2
L 
b2b1
b
b2b3 
b2
1 2

1  b0
1  b0
1  b0 


2
b3b1
b3b2
b3 


b
1

 3
1  b0
1  b0
1  b0 

b is the boosted four-velocity.
 E 
 p 
Four-momentum:  x   L
 py 
 
 pz 
E
p 
 x  Four-coordinate:
 py 
 
 pz 
 t 
 x 
 L
 y 
 
 z 
t
 x
 
 y
 
z
Wigner phase-space density for Deuteron
Wigner transformation
dW ( r , k )   d 3 Re ik r  ( r R / 2)  ( r  R / 2)
Hulthen wave function
 (   ) e r  e  r 15  2i   r
 (r) 
  ci 
 e
2
2 (   )
r
  
i 1
3/ 4
2
i
4
0.4
0.3
3
0.2
2
 (k)
 (r)
Hulthen
Hulthen wih 15 Gaussians
0.1
0.0
  0.23 fm 1
  1.61 fm 1
r 2  1.89 fm
1
0
2
4
6
r (fm)
8
10
12
0
0.0
0.5
1.0
1.5
2.0
k (1/fm)
Chen,Ko,Li, NPA729
2.5
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.,
1
 m 
 (r)  
exp(

m r 2 )

2
 2 
with  the harmonic oscillator frequency
3/ 4
t/3He Wigner phase-space density and root-mean-square radius:
 t/W He (ρ, λ; k  , k  )  82 exp(   2 /  12   2 /  22   12 k 2   22 k2 )
3
2
t/ 3 He
r
1 m12 (m2  m3 )  m22 (m3  m1 )  m32 (m1  m2 )

(t: 1.61 fm; 3 He: 1.74 fm)
2
( m1  m2  m3 )m1m2m3
ρ
1
3
m1
m2
(r1  r2 ), λ 
(
r1 
r2  r3 ) (Jacobi Transformation)
2
m

m
m

m
2
1
2
1
2
k 
2
6
( m2k 1  m1k 2 ), k  
( m3k 1  m3k 2  ( m1  m2 )k 2 )
m1  m2
2(m1  m2  m3 )
 12  ( 1 ) 1 and  22  ( 2 ) 1 with
 1
1 
1  2   
 m1 m2 
1
3 1
1 
and 2  


2  m1  m2 m3 
1
Isospin symmetric collisions at E/A≈100 MeV
36
58
o
Ar+ Ni@E/A=95 MeV, 60 <c.m.<120
-1
10
data (b=4-5 fm)
IBUU+Coalescence
(b=4.5 fm)
-2
10
o
data (b=6-7 fm)
IBUU+Coalescence
(b=6.5 fm)
-3
10
Try Coalescence model
at intermediate energies!
-4
10
-5
(a) Deuteron
(d) Deuteron
(b) Triton
(e) Triton
 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
10
-2
-1
dM/dEkin (MeV )
10
-3
10
-4
10
-5
10
-6
10
-7
10
-2
10
-3
10
-4
10
-5
10
-6
10
3
10
3
(c) He
-7
0
50
(f) He
100
150
200
250
0
50
Ekin (MeV)
100
150
200
Data are taken from INDRA
Collaboration (P. Pawlowski, EPJA9)
250
300
Chen,Ko,Li, NPA729
Symmetry Energy Effects on t/3He ratio
2.5
52
48
3
Y(t)/Y( He)
Ca+ Ca, E=80 AMeV, b=0 fm
=0.5
=2.0
=0.5 with soft EOS
=0.5 with medium
2.0
1.5
50
100
150
200
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+B
X: isospin tracer
E=50 AMeV and b=6 fm
 0 (Stronger);
 1 (Weaker)
_____________
IBUU04
Chen,Ko,Li,
PRL94,2005
MDI interaction
Lane Potential
Chen,Ko,Li, PRL93,2005
MDI ~Finite Range Gogny Interaction
GMR (Shlomo &Youngblood,PRC47 ):
Kasy : 566  1350 34  159MeV)
Discussions
1. Effects of momentum-dependence of nuclear potential
Two-nucleon correlation functions
8
(a) nn
P<300 MeV/c
7
6
52
(d) nn
P>500 MeV/c
MDI
Das, Das Gupta, Gale and Li
PRC67, (2003)
48
Ca+ Ca
E=80 AMeV, b=0 fm
5
4
3
2
Stiff Symmetry Energy: MDI with x  2
Soft Symmetry Energy: MDI with x  1
1
(b) pp
C(q)
1.5
(e) pp
1.0
Pairs with P>500 MeV:
n-p CF: 11%
0.5
The sensitivity becomes weaker
with momentum-dependence
0.0
(c) np
4.0
3.5
(f) np
MDI with soft sym. pot.
MDI with hard sym. pot.
3.0
2.5
2.0
1.5
1.0
10
20
30
40
10
q (MeV/c)
20
30
40
50
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
2. Effects of momentum-dependence of nuclear potential
t/3He ratio
2.5
(a) SBKD
(b) MDI
52
3
Y(t)/Y( He)
Stiff Symmetry Energy:
MDI with x  2
Soft Symmetry Energy:
MDI with x  1
48
Ca+ Ca
E=80 AMeV, b=0 fm
Soft Sym. Pot.
Hard Sym. Pot.
2.0
1.5
0
20
40
60
80 100 0
20
Ek (MeV)
40
60
80 100 120
Still sensitive to the stiffness of the symmetry energy
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
Li,Chen, Nucl-th/0508024.
Ri(isospin transport/diffusion)
Symmetry potential and np collisions
 The  parameter is found to be between 0.69 and 1.05
 The K asy is norrowed down to  500  50 MeV, which agrees very well with
the giant resonance results about Sn isotopes (by Fujiwara)
 Compared with the experimental data about the n-skin of
208
Pb:   0.8
4. Have We Already Known the Density Dependence of
Nuclear Matter Symmetry Energy at Sub-saturated
Densities?
arXiv:nucl-ex/0505011
Isocaling+AMD
_______________
________________________________
___________________
Esym (  )=31.6( /0 )
W. D. Tian, Y. G. Ma, et al.,  Isospin Transport/Diffusion:   0.69  1.05
Isoscaling + CQMD
 Isoscaling+AMD:   0.6  1.05
 Neutron-skin of 208 Pb:   0.8
  0.7 is most acceptable
5. The High Density Behaviors of Nuclear Matter Symmetry
B. A. Li, PRL88 (2002) 192701
Li,Chen,Ko,Yong,Zuo, nucl-th/0504008;
Li,Chen,Das, Das Gupta,Gale,Ko,Yong,
Zuo, nucl-th/0504069
Other possible observations: Kaons, Σ, …
nucl-th/0504065, Phys.Rev. C71 (2005) 054907
————————————————————
—————————————————————————
——————————————
6. Momentum Dependence of Symmetry Potential
Di Toro et al.
Recent progress:
‘Puzzle’?
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
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！