Effect of Friction on Neutron Emission in
Fission of Heavy nuclei
Jing-Dong Bao, Ying Jia
Department of Physics, Beijing Normal University
May 16-12, 2006 Shanghai, China
1. The parameterization
2. Stochastic calculation for angular distribution
of fission fragment
3. Effect of friction on neutron emission
4. Summary
1.{c, h, α}parameterization of nuclear
shape
2
2
2
2
2
2


z ,
c


Bz

c
A
)
z

c
(
c

s
2
 s ( z )   2 2
c ( c  z 2 )As c 2  c 2 z exp Bcz 2 ,
B  2h 
B  0,
B  0,
c 1
2
c 3  B / 5,

As   4
B

 3 exp( Bc 3 )  1  1 2 Bc 3   Bc 3 erf (  Bc 3 ) ,



B0
B0
where c is the elongation, i.e., the distance of two centers of
fragments ; h is the neck parameter; αis the asymmetrical parameter.
A set of Langevin equations
1
ij
qi (t )  m p j
test particles: 10
5
V ( q, N ) 1
1 m jk ( q)
p i (t )  
 mik ( q) kj ( q) p j (t ) 
p j pk  gij ( q) j (t )
qi
2 qi
i (t )  0,
i (t )j (t ' )  2 ij (t  t ' )
.
2

Z2
 N  Z   2/3
2 5 / 3
V (q, N )  a 2 1   
  A Bs (q)  1  c3 1 / 3 Bc (q)  1  cr L A Br (q)
A
 A  

E*  Etot  V  Ecoll  En
T 
E* a
Stochastic Runge-Kutta algorithm
t
q(t  t )  q(t ) 
2
 p(t )
p * (t ) 
 m(q)  m(q*) 


t
1
p(t  t )  p(t )  hq(t ), p(t )   hq * (t ), p * (t )   g q(t )   g q * (t )  2t
2
2
V (q)  (q)
1 m(q) p 2 (t )
hq(t ), p(t )   

p(t ) 
q
m(q)
2 q m 2 (q)
g (q)  k BT (q)
.
Time-dependent fission rate （5  105） 254
Cf
100
(a)
-1
r (10 sec )
80
60
-1
17

40
-1

(x0->xb)=63.1
mfpt
LE=28.58
(x0->xb)=26.4
mlpt
20
passing
over the
scission
test particles passing
over the saddle point
first time
-1

0
0
100
(x0->xsc)=23.1
mfpt
200
t (10
-21
test particles passing
over the saddle point
multi-times
300
sec)
J. D. Bao, Y. Jia: PRC 69, 027602 (2004).
400
*During the Monte-Carlo simulating step, we compare
time step 
and half decay period of neutron  n (  / n )
 /  n  Rn
Rn
is a random number between 0 and 1，then emission
of a neutron;
* The nuclear inertial energy decreases with the
emission of neutron。
5.0
1D
2D
Experimental data
Average Neutron Numbers
4.5
4.0
3.5
3.0
2.5
2.0
16
1.5
208
224
O+
Pb--->
Th
130
140
150
1.0
90
100
110
120
ELab/MeV
Angular distribution of fission fragment


( 2 I  1) exp  p sin 2  J 0 (  p sin 2  )
W ( , I ) 
,
erf ( 2 p )
p  ( I  1 2) 2 /( 4 K 02 )
K 
2
0
J eff

A
2
Tsd ,
1
J eff
 J ||1  J 1
W (0 0 )
W ( 90 0 )
Effective rotational
moment defined at
saddle-point and is
always a random
variable!
Tsd is the temperature at the saddle point, calculated
by Langevin simulation
4.8
1D
2D
Experimental data
4.0
3.6
0
0
A=W(0 )/W(90 )
4.4
3.2
2.8
16
2.4
208
224
O+ Pb---> Th
2.0
90
100
110
ELab/MeV
120
130
3. One-body well plus window friction
 wall
z max  P
P z c 
1
s


 k s m v  


z min
2

c

z

c


2
1
 R 
  m v
 
2
 c 
2
s
2
2



P
1
s
  Ps2  

 2 z




2




1 / 2
2
 win

is window area
 ww   wall   win
The well formula is applied near the ground state;
the window formula is addressed when the system arrives at
the saddle point. In general, we have:
 f  f w  (1  f ) ww
dz
1400
w
fw
ww
gw
obd
1200
-21
/(10 Mev s)
1000
800
600
400
200
0
0.4
0.6
q/R0
0.8
1.0
1.2
5.0
Nexpe
Npresci
Nf
Nl
4.5
N
4.0
3.5
3.0
2.5
2.0
1.5
90
100
110
120
130
ELab/MeV
Neutron emission pre-scission for Th224
Other questions:
•The two-dimensional fission rate is (20-30%) larger
than that of one dimension if the neck is considered;
•Dependence of Transport coefficients on
temperature needs to consider in further work.
7
Nneu=0
Nneu=2
6
5
V(q)
4
3
2
1
0
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
q
The saddle-point drifts into inside well and the fission
barrier decreases with the emission of neutron
4. Summary
(1)The limitation of diffusion model：it is not suitable to
quasi-fission problem, because an equilibrium shape is necessary.
(2)Underdamped motion before saddle-point and overdamped
from saddle-point to scission point;
(3) Temperature at saddle-point determined by the case of test
particle leaving saddle point last time;
(4)Oscillating time around saddle-point is the longest dynamical
time scale, but it is influenced strongly by neutron emission.
Thanks !