Symmetry energy and pion production in
the Boltzmann-Langevin approach
Wen-Jie Xie, Jun Su, Long Zhu, Feng-Shou Zhang
谢文杰，苏军，祝龙，张丰收
Supervisor: Prof. Feng-Shou Zhang
Beijing Normal University
北京师范大学核科学与技术学院
2012.08
Contents
1. Introduction
2.The isospin- and momentum-dependent Boltzmann-Langevin Model
(IBL)
3. The calculated results on the pion production using the IBL model
(1) the pion multiplicity


(2) the   ratio
4. Conclusions
Introduction
The form of symmetry energy is important in both nuclear physics
and astrophysics and not well constrained up to now.
At subnormal densities:
The symmetry energy increases with increasing the density.
At supernormal densities:
The trend of the symmetry energy with increasing density is not constrained.
Supersoft : IBUU04, Z. G. Xiao, PRL102 (2009) 062502
Superstiff: ImIQMD, Z. Q. Feng, PLB 683 (2010) 140
Among the probes of the symmetry energy, the     is promising.
The resonance model predicts a ratio of

where the
N Z



 5N  NZ
2
 5Z
2
 NZ    N Z

2
is determined by the symmetry energy.
R. Stock, Phys. Rep. 135 (1986) 259
,
The IBL model
1. The Boltzmann-Langevin equation :
p
 

  r   rU

 t m
 fˆ   
p
 ˆ
 f  r , p, t   K

 fˆ    K  r , p , t 
2. The evolution equation :
3. The effective potential :
V  V sky 2  V sky 3  V sym  V m di  V coul
4. The collision term:
5. The fluctuation term:
Qˆ 2 0  r , t   t   Q 2 0  r , t   t  
 tC 2 0  r , t W 2 ,
Qˆ 3 0  r , t   t   Q 3 0  r , t   t  
 tC 3 0  r , t W 3 .
The fluctuation term
The fluctuating collision term  K  r , p , t  can be interpreted as a
stochastic force acting on f  r , p , t   t and is charaterized by a
correlation funtion,
 K  r1 , p1 , t1   K  r2 , p 2 , t 2   C  p1 , p 2    r1  r2    t1  t 2  .
After using an approximate treatment, the local momentum
distribution is projected on a set of low-order multipole moments Q L M
of order L with magnetic quantum numbers M . And the C  p1 , p 2 
can be reduced and written as:
C L M L M   r , t  

 dp dp
1
2
 dpdp Q  p  Q
LM
L M 
 p C  p, p
dp 3 dp 4  Q L M  Q L M   W  12, 34  f 1 f 2 1  f 3  1  f 4  ,
F. S. Zhang et al. Phys. Rev. C 51(1995) 3201
E. Suraud et al. Nucl. Phys. A542 (1992) 141
Y. ABE, S. Ayik, et, al. Phys. Rep. 275 (1996)49
The scaling procedure
The scaling procedure used to rescale the local momentum distribution to the local
values of Qˆ 2 0  r , t  and Qˆ 3 0  r , t  is given in the following
pz  0 : p z   p z , p    p ;
pz  0 : pz   pz , p    p .
The  ,   ,   are solved in terms of the following equations:
2mE  

2
2

pz   
2
2
2
2
Q 30  2
3

p   
2
2
pz 0
pz   
Q 20  2
2
2

pz   
2
2
 
2
p

p z  3  
3
2
2
p
pz 0
2
p

2
p

 
2

2
p
2
px p
,

,
 
2

F. S. Zhang et al. Phys. Rev. C 51(1995) 3201
2
p z p

.
The evolution process of IBL
1. Starting with a definite density f  r , p , t  at time t , the first step
is to determine the local average evolution from t to t   t , which
yields f  r , p , t   t  and the elements of the diffusion
matrix C L L   r , t  .
2. The diffusion matrix is diagonalized and the fluctuations are
calculated.
3. The fluctuations are inserted into the single-particle density. The
above three steps are repeated at each time step.
The differences between the IBL and usual BUU models
Vlasov: 演化过程只受哈密顿
方程的约束，每次事件模拟
的状态确定，不需做多次事
件模拟。
BUU： Vlasov+ 随机碰撞，尽
管碰撞具有随机性，但平均
路径是一样的，只能讨论单
体观测量。
BLE：BUU+ 涨落，涨落的效
果是在每一步演化的过程中
加一随机量，使得系统每次
演化的路径不一样，使得模
型可以讨论除单体观测量意
外的其他观测量
A. Ono, J. Randrup, Eur. Phys. J. A 30 (2006) 109
The differences between the IBL and usual BUU models
500
500
400
Q20+20
300
-2
Q20-20
-2
Q20 (fm )
300
Q20 (fm )
BUU： 动量四极距是确定的
BLE：由于涨落的存在，动量四极
距值有较大的变化范围
Q20
400
(a)
Q20+220
200
Q20-220
100
200
Q20
0
0
20
Q20+20
100
40
60
t (fm/c)
Q20-20
0
Q20+220
Q20-220
-100
50
(b)
-2
20 (fm )
40
IBL
BUU
30
20
10
0
0
10
20
30
40
50
60
t (fm/c)
The  20

Q 20  Q 20
2
2
is the standard deviation function.
The forms of the symmetry energy
100
80
Esym (MeV)
  
  
E sym (  )  38.9    18.4  
 0 
 0 
soft
linear
hard
supersoft
  
E sym (  )  14.7  
 0 
60
40
20
0
0
1
2
3
4
/0
Z. Q. Feng, G. M. Jin, Phys. Lett. B 683 (2010) 140
s
2.14
  
 3.8  
 0 
5/3
Pion multiplicity
100
100
10
M()
M()
10
FOPI data
IBL SM
IBL HM
1
1
0.1
0.2
0.4
0.6
0.8
1.0
FOPI data
supersoft
soft
linear
hard
0.2
1.2
Ein (GeV/nucleon)
100
M()
10
FOPI data
IBL
BUU
Xiao PRL09
1
0.1
0.2
0.4
0.6
0.8
1.0
1.2
Ein (GeV/nucleon)
W. Reisdorf, et, al., Nucl. Phys. A 781(2007) 459
Z. G. Xiao, et, al. ,Phys. Rev. Lett. 102 (2009) 062502
0.4
0.6
0.8
1.0
Ein (GeV/nucleon)
1.2
The dependence of the 


 ratio on the N/Z at 400A MeV
4.0
3.5
FOPI data
supersoft
soft
linear
hard
3.0
3.0
96
197
Au+ Au
96
Zr+ Zr
2.5
(N/Z)
+
 /
2.0
-
-
 /
+
2.5
197
FOPI data
IBL
BUU
3.5
2.0
1.5
1.5
96
40
2
96
Ru+ Ru
40
N/Z
Ca+ Ca
1.0
1.0
1.0
1.2
1.4
1.6
0.5
1.0
N/Z
W. Reisdorf, et, al, Nucl. Phys. A 781(2007) 459
R. Stock, Phys. Rep. 135 (1986) 259
1.2
N/Z
1.4
1.6
The excitation function of
5



5
FOPI data
supersoft
soft
linear
hard
4
 /
+
+
3
-
-
from central Au+Au collisions
FOPI data
IBL
BUU
4
 /

2
0.0
3
2
0.2
0.4
0.6
0.8
1.0
Ein (GeV/nucleon)
1.2
1.4
0.0
0.2
0.4
0.6
0.8
1.0
Ein (GeV/nucleon)
1.2
1.4
Conclusions
1.
2.
3.
The pion multiplicity is dependent on the EOS and independent
on the symmetry energy.
The pion multiplicity and     ratio calculated by the IBL
mdoel are larger than those obtained by the usual BUU model
at lower incident energies, especially for below the pion
threshold value.
Calculations with a supersoft symmetry energy describe well
the experimental data.
Thank you for your
attention！