Journal of Applied Mathematics and Physics, 2014, 2, 27-31
Published Online April 2014 in SciRes. http://www.scirp.org/journal/jamp
http://dx.doi.org/10.4236/jamp.2014.25004
How to cite this paper: Gontchar, I.I., et al. (2014) Dynamical Modeling of the Nuclear Fission Process at Low Excitation En-
ergies. Journal of Applied Mathematics and Physics, 2, 27-31. http://dx.doi.org/10.4236/jamp.2014.25004
Dynamical Modeling of the Nuclear Fission
Process at Low Excitation Energies
I. I. Gontchar1, M. V. Chushnyakova1,2, E. P. Oskin1, E. G. Demina1
1Physics and Chemistry Department, Omsk State Transport University, Omsk, Russia
2Physics Department, Omsk State Technical University, Omsk, Russia
Received December 2013
Abstract
Two recipes for modeling the dynamics of the nuclear fission process are known in literature. The
underlying equations contain the driving, dissipative, and random forces. The two recipes are
mostly different in the prescriptions for the driving force. In this work we carefully compare these
driving forces and the resulting fission rates. It turns out that the rates may be very close or
strongly different depending on the value the shell correction to the nuclear deformation energy.
We give arguments in favor of one of the recipes.
Keywords
Nuclear Fissio n, Dynamical Modeling, Stochastic Differential Equations, Fermi Gas Model
1. Introduction
Numerous experiments related to the nuclear fission process [1,2], in particular those aiming for discovering
new superheavy elements [3-5], require practical and reliable numerical modeling of fission dynamics at low
excitation energies. The most important physical quantity characterizing the fission process, although not ob-
servable, is the fission rate which is by definition
( )
1
()( )
1
()
fi fi
i
totf i
Nt Nt
Rt N Ntt
−
−
=−∆
, (1)
1
() ()
i
fif j
j
Nt Nt
=
= ∆
∑
. (2)
Here
( )
fj
Nt∆
is the number of trajectories reached the scission point during the time interval
1jj
tt t
−
∆= −
;
( )
fi
Nt
is the number of trajectories reached the scission point until the time moment
i
t
.
Presently the dynamical modeling of fission is mostly performed by solving numerically the stochastic differ-
rential equations with the white noise. In the simplest one dimensional case these equations read
1
dqpm dt
−
=
(3 )
1
2dpm pdtfdtT dW ηη
−
=−++ ⋅
(4)

I. I. Gontchar et al.
28
Here
dt
is the time interval during which the collective coordinate (deformation parameter)
q
and its con-
jugate momentum
p
increase by
dq
and
dp
, respectively;
W
is the Wiener process whose increment
dW
obeys the normal distribution with the variance
dt
(
~dW dt
);
η
and
m
are the friction and inertia
parameters, respectively. For the sake of simplicity we consider
η
and
m
to be deformation independent
(
/m βη =
).
The driving force can be calculated by two methods. The first one is a generalization [6] of the thermody-
namical approach, which was developed for the one dimensional problem in [7,8] and later was extended to the
multidimensional case in [9]. Note that this method is applied, except [6], for the high excitation energies when
the shell correction to the nuclear Potential Energy (
V
, PE) and to the single particle Level Density Parameter
(
a
, LDP) is believed does not play a role. The driving force calculated using this approach,
I
f
, is defined as a
proper derivative of a thermodynamical potential:
I
TE
FS
fT
qq
 
∂∂
=−=
 
∂∂
 
(5)
is the Helmholtz free energy,
( )
,S qE
is the nucleus entropy, and
E
is its total excitation energy. Within the
framework of the Fermi gas model, the entropy reads
()()( )
1/2
,2,SqEaqU Uq=

(6)
The intrinsic excitation energy
U
is related to
E
and PE as follows:
()( )( )()
LS
UqEVqEVq Vq=−=− −
(7)
Here
( )
L
Vq
and
( )
S
Vq
are the smooth (liquid drop) part of PE and the shell correction, respectively.
According to [10] the LDP can be written as
()( )()( )
{ }
1
,1
LS
aqUaqgUV qU
−
= +
(8)
The function
( )
gU
reaches unity when
U
becomes large enough:
()( )
1 exp g UkU=−−
(9)
1
0.054 MeVk
−
=
. The deformation dependence of the smooth part of the LDP is defined as follows [10]:
( )( )
2/3
12 Ls
aq aAaABq= +
. Here
A
is the nucleus mass number,
1
1
0.073 MeVa
−
=
and
1
20.095 MeVa−
=
,
( )
s
Bq
is the dimensionless nuclear surface area.
Using Equations (5), (7) and (8) we obtain for the driving force the following expression
2
I LSE
fV VaT
′′′
=−− +
(10 )
The prime denotes the derivative with respect to the coordinate, the subscript
""E
indicates that the deriva-
tive must be taken keeping
E
constant.
Within the framework of the second approach [11,12] the driving force
II
f
is calculated according the for-
mula, which (using our notations) reads
( )
1
II LS
f VgV
′′
=−−−
(11)
Equations (10) and (11) look very much different. One hardly can expect to find the fission rates similar if
I
f
and
II
f
are used in Equation (4). In order to check whether the rates are different indeed we construct a sche-
matic PE which reproduces the main distinct features of the PE of 236U-nucleus. This schematic
( )
Vq
is shown
in Figure 1 along with
( )
L
Vq
and
( )
S
Vq
.
One sees clearly the double humped structure of the fission barrier which is well known for the actinide nuclei
[13].
We modeled the fission process applying Equations (3), (4) and the driving forces
I
f
and
II
f
. The PE of Fig-
ure 1 was used for the modeling. The initial conditions corresponded to the Brownian particles at rest at the left
minimum of the PE. Typical fission rates resulted from this modeling according to Equation (1) are shown in
Figure 2. Although the quasistationary rate
DII
R
exceeds
DI
R
by some 20%, one can expect much larger dif-
ference considering Equations (10) and (11).
In order to figure out the reason for this surprisingly small difference let us transform Equation (10). The de-
rivative of the LDP reads:

I. I. Gontchar et al.
29
Figure 1. The schematic PE for 236U (curve
with dots) as a function of collective coordinate
along with its components: the smooth (solid
curve) and shell correction (dashed curve) parts.
Figure 2. The dynamical fission rates calcu-
lated for 236U according to Equation (1) versus
modeling time. The horizontal lines indicate
the quasistationary values of the rates.
2
SSS S
LL
E
ag VgVgVUgV
aa
qUU UU
 ′′′  ∂⋅ ⋅ ⋅⋅⋅
′
= +−+
 
∂
 
( 12)
Keeping in mind that
LS
U VV ′ ′′ =−−
and
( )
expg kUkU
′′
= −
we convert Equation (10) to the form
1
SS
IL
UV kVgVV
fV akV
U UU
ε
εε
ε
′ ′′
⋅

′′ ′
=−+ +−+−

+
(1 3)
Here
()()
1
S
gUV qU ε
−
=
turns out to be a small parameter not exceeding 0.1 in the region of the barrier at
10U>
MeV in particular because in the considered case
( )
S
Vq
is in order of 2 MeV. Omitting in (13) the
terms obviously proportional to
ε
we arrive at the formula
IIIS LS SL
ffVkVV kVgV ′ ′′ = +++⋅
(14)
Since
1
k
−
is about 20 MeV, in Equation (14) the terms containing
( )
S
kV q
are rather small. The last term is
small because
( )
L
Vq
is rather smooth (see Figure 1). We believe that this derivation explains relatively small
difference between
I
f
and
II
f
resulting in small difference between the corresponding fission rates in Figure 2.
We also have considered the case of a nucleus with the large shell correction at the ground state, namely the
double magic lead-208. The PE of Figure 3 was used for the modeling in this case. The absolute value of the
shell correction is now about 12 MeV. The way of modeling Figure 4. One sees that in this case
DII
R
exceeds
DI
R
by factor of 2 whereas the conventional value of the intrinsic excitation energy at which the shell correction
ceases to manifest itself (and consequently
DII
R
and
DI
R
must converge) is about 60 MeV.
It is worthwhile to discuss and illustrate what does the smearing of the shell correction exactly mean. Within
the framework of the first recipe, it turns out that one can calculate the fission rate ignoring completely the shell

I. I. Gontchar et al.
30
correction in the PE (i.e. using
L
V
instead of
V
in Equation (7)) and in the LDP (i. e. using
L
a
instead of
a
in Equation (6)). The corresponding fission rate
DIL
R
must converge with
DI
R
as
E
increases. This is seen in
Figure 5 where the quasistationary rates resulting from dynamical modeling are shown. In order to quantify the
convergence between
DIL
R
and
DI
R
we have calculated the fractional difference
/1
DI DIL
RR−
. It has turned out
that the difference goes to zero reaching circa −10% at
60E=
MeV as it is known for the statistical rates.
0.2
0.4
0.6
0.8
1.0
1.2
Figure 3. The schematic PE for 208Pb (curve
with dots) as a function of collective coordi-
nate along with its components: the smooth
(solid curve) and shell correction (dashed
curve) parts.
0.25
0.20
0.15
0.10
0.05
0.00
Figure 4. The dynamical fission rates calcu-
lated for 208Pb according to Equation (1) ver-
sus modeling time. The horizontal lines indi-
cate the quasistationary values of the rates.
Figure 5. The dynamical quasistationary fis-
sion rates calculated for 208Pb nucleus ac-
counting for the shell correction (squares) and
ignoring those (line with circles).

I. I. Gontchar et al.
31
Within the framework of the second recipe, the convergence is not reached because the reference point for the
excitation energy always include the potential energy with the shell correction (see Equation (6) of [12]).
To finalize, we have compared two approaches for calculating the driving force for the nuclear fission process
at low excitation energies when the shell effects are expected to be significant. We have found that in the case of
uranium-236 nucleus the quasistationary decay rates
DI
R
and
DII
R
resulting from these approaches are rather
close (the difference is about 20%). This is however just because for this nucleus the shell correction is small in
comparison with the typical energy
1
18.5k
−
=
MeV controlling the smearing out the shell effects. For the lead-
208 nucleus with larger value of the shell correction, the difference between
DI
R
and
DII
R
reaches factor of 2.
This is significantly larger than the difference between the rates calculated within the frame work of the first ap-
proach with and without the shell correction. Since the first approach is based on the thermodynamical argu-
ments, we are inclined to make favor to it in comparison with the second one.
Acknowledgements
M. V. C. and E. G. D. are grateful to the Dmitry Zimin Foundation ‘Dynasty’ for financial support.
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