determination of density dependence of nuclear matter symmetry energy in hic’s
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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
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!
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