Journal of Nuclear and Radiochemical Sciences,Vol. 3, No. 1, pp. 67–69, 2002

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Dynamics of Fission Modes Studied with the 3-dimensional Langevin Equation
T. Ichikawa,∗ T. Asano, T. Wada, and M. Ohta
Department of Physics, Konan University, Kobe 658-8501, Japan
Received: December 26, 2001; In Final Form: April 18, 2002
We investigate fission modes of 270 Sg with 3-dimensional Langevin equation. The shell correction energy is includedin the total energy. The mass distribution and the total kinetic energy (TKE) distribution of fission fragments arecalculated and are compared with the experimental results. Mass asymmetric fission dominates in the mode withthe lower TKE, which is in agreement with the experimental result. The dynamics from saddle to scission is alsodiscussed.
1. Introduction

dqi= m−1 i j p j ,dt∂V 1 ∂  −1 d pi−m jk p j pk= −dt∂qi 2 ∂qi
− γi j m−1 jk pk + gi j R j (t),(1)

where the suffix stands for Z0 , δ, or α. Summation over repeatedindices is implied. V (q) is the potential energy taken accountof shell effect, mi j (q) and γi j (q) are the shape dependent collective inertia and dissipation tensors. We assume the randomforces as the white noise type of which the normalized randomforce R(t) is to satisfy R(t) = 0, Ri (t1 )R j (t2 ) = 2δi j δ(t1 − t2 ).
The strength of random force gi j is calculated from γi j T = gik gk jthat is given by the fluctuation-dissipative theorem. T denotes
the temperature of the compound nucleus that is defined asE ∗ = aT 2 with the excitation energy of compound nucleus andthe level density parameter a of Töke and Swiatecki.4 The inertia tensor is calculated using the hydrodynamical model with
Werner-Wheeler approximation5 for the velocity field, and walland-window one-body dissipation6 is adopted for the dissipation
tensor.
The shell correction energy of the two-center shell model iscalculated with the code TWOCTR.7, 8 The shell correction energy depends on the temperature of the nucleus. The temperature dependent factor of the shell correction energy is assumed
as exp (−E ∗ /Ed ), where E ∗ is the excitation energy and Ed is

δIn the heavy compound nucleus with A > 200, fission modes
have been investigated from the measurement of the mass andthe total kinetic energy (TKE) distribution at low excitation energy and the effect of shell energy are established well. At low
excitation energy, the asymmetric fission mode is observed for
all nuclei with A > 200.
For example, the measurement of the mass-energy distribution of the fission fragments of 270 Sg (Z = 106) was performedby Itkis et al.1 It was observed that at low excitation energy,
the mass asymmetric fission dominates in the components ofthe mode having lower TKE, whereas this tendency vanishes athigh excitation case. The domination of the mass asymmetricfission in the components having low TKE is the characteristicof the nuclei heavier than the fermium region. From these results, the existence of the different dynamical fission paths waspointed out.
The problem of fission modes has also been studied theoretically. For example, Pashkevich calculated the energy surface inthe multi-dimensional parameter space including the shell energy and found several saddle points and fission valleys.2 However, it was a static calculation and was not sufficient to understand the mass-energy distribution. In order to understand these
phenomena, the effect of the shell correction energy to the dynamical fission path is essential.
Therefore, we investigate fission modes from the mass andTKE distribution calculated with the 3-dimensional Langevinequation and investigate the influence of dynamics from saddle
to scission. At the high excitation energy where the shell effectvanishes, the fission process has been studied on the basis of
the fluctuation-dissipation dynamics and the Langevin equation3
has been used to describe this dynamics. The 3-dimensional
Langevin calculation agreed with experimental values of themass distribution and the total kinetic energy of fission fragments. In this study, including the shell correction energy, weextend the 3-dimensional Langevin calculation to the case of
the low excitation.
In this paper, we investigate fission modes of 270 Sg systemand present results of the calculation of the mass distributions
and the total kinetic energy of fission fragments. The influenceof the potential energy on the dynamics is also discussed.

same deformation (δ1 = δ2 ), and α denotes the mass asymmetry parameter (α = (A1 − A2 )/(A1 + A2 )), where A1 and A2 are
the mass number of the fragments. The liquid drop energy, thesurface energy, and the coulomb energy are also calculated withthis parametrization.
We describe the fission process by the following equation,
called as the Langevin equation,

2. Framework

The shape of nucleus is described by the Two-Centerparametrization and Figure 1 shows various shapes of 270 Sg withthis parametrization. Z0 denotes the distance between the harmonic oscillators in the unit of the radius of the spherical compound nuclei (R0 = r0 (A1 + A2 )1/3 ), δ denotes the deformationof the fragments with the constraint that both fragments have∗Corresponding author. E-mail: dn021001@center.konan-u.ac.jp.

FAX: +81-78-435-2539.

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Figure 1. The various shapes with the Two-Center parametrization inthe Z0 -δ plane.

c 2002 The Japan Society of Nuclear and Radiochemical SciencesPublished on Web 6/30/2002

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J.Nucl.Radiochem.Sci.,Vol. 3, No. 1, 2002

Ichikawa

Figure 2. The landscape of this figure shows the total energy projectedto Z0 -δ plane of 270 Sg. The circle denotes the ground state of compound
nucleus and the cross denotes the first saddle point in the case of α = 0.
The line with dot is the sample trajectory of the calculation.



Figure 4. The distribution of the total kinetic energy of fission fragments. The dashed line denotes the TKE distribution for δ > 0.3. The
dotted line denotes the case of δ < 0.3.

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δ

Figure 3. The distribution of the deformation of fission fragments atscission.

the shell damping energy that is taken to be 20 MeV.9
Figure 2 shows the potential landscape of 270 Sg at E ∗ =10 MeV in the case of α = 0. The circle denotes the groundstate of this system and is the start point of the Brownian particle in Langevin calculation. The cross at Z0 = 0.3 and δ = 0.2 inFigure 2 indicates the first saddle of this system in the case ofα = 0.
3. Numerical Result
We investigate fission modes of the compound nucleus
Sg in the case that the excitation energy E ∗ is 10 MeV(T ∼ 0.6 MeV). In this system, the fission barrier height B f isapproximately 5 MeV.
The histogram in Figure 3 shows the distribution of the deformation of fission fragments. In this figure, two peaks can beseen clearly at δ = 0.25 and 0.36. From this result, we found thatdifferent dynamical fission paths exist.
The histogram in Figure 4 shows the distribution of the TKEof fission fragments. In this figure, different from Figure 3, it
is not apparent that this distribution consists of several components. However, it is found that this TKE distribution cannot befitted by a single Gaussian. According to the result that is shownin Figure 3, we divide the events of the fission fragments into
two parts: one is the events with the deformation of fragments
being δ > 0.3 and the other is that with δ < 0.3. In Figure 4,
the dashed line denotes the TKE distribution of the events with
δ > 0.3 and the dotted line denotes that with δ < 0.3. We foundthat the TKE of fragments with δ > 0.3 is smaller than that with270Figure 5. The distribution of the mass asymmetry of fission fragmentswith the deformation δ > 0.3. The dashed line denotes the one fittedwith four Gaussians and the dotted lines denote the main componentsof the fit.

δ < 0.3. Fragments with lower TKE have larger deformation.
It is seen that the TKE distribution with δ > 0.3 has a simple
peak structure and can be expressed as a single Gaussian. We
expect that the distribution with δ > 0.3 is from a single mode.
As for components with δ < 0.3, the statistical frequency is low
compared with that with δ > 0.3. We expect from Figure 3 thatthe events with δ < 0.3 contain the contribution from the distribution that has a peak around δ ∼ 0.36. Since we still cannot
extract a single mode from the components with δ < 0.3, we investigate the mass distribution only for the events with δ > 0.3
in this study.
The histogram of Figure 5 shows the distribution of the mass
asymmetry α of fission fragments with δ > 0.3. The dashed lineis the result of the fitting with four Gaussians and the dottedlines are the main Gaussian components that are used in the fitting. From this result, the asymmetric fission can be seen clearly.
Peaks of the main Gaussian components are at α = 0.056 and−0.057.
The experimental results show that the mass asymmetric fission dominates in the mode having lower TKE distribution. Itkis
et al.1 proposed that the reaction of 270 Sg → 127 Sn + 143 Ba corresponding to Standard II mode was the main fission channel dueto the effect of the deformed shell (Z = 56) and this reaction corresponds to α = 0.06. We consider that our calculation resultscorrespond to this experimental one. However, the peak energyof the TKE distribution in our calculation is larger than that ofthe experiment by about 30 MeV. At present stage, this difference is yet to be clarified.
Next, we consider the origin of this mass asymmetric dis-Dynamics of Fission ModesJ.Nucl.Radiochem.Sci.,Vol. 3, No. 1, 2002

α          δ

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Figure 6. The energy landscape of 270 Sg projected to Z0 -α in the caseof the deformation δ = 0.2.

Figure 7. The energy landscape of 270 Sg projected to Z0 -α in the caseof the deformation δ = 0.36.

tribution. Although the dynamical motion is calculated in thethree-dimensional parameter space, it is instructive to look at thepotential surface that are projected onto two-dimensional spaces
like Z0 -α and Z0 -δ. Figure 6 shows the potential projected toZ0 -α at δ = 0.2 that corresponds to the saddle point deformation in Z0 -δ plane of Figure 1. Figure 7 shows the potential
projected to Z0 -α at δ = 0.36 that corresponds to the maximumof the distribution in Figure 3. In the Z0 -α plane, the saddle
point locates around at Z0 = 0.5 and α = ±0.15. First, in Figure 6, when fissioning nucleus goes from the ground state to thesaddle point according to potential surface, the mass asymmetry α increases (the arrow 1). After passing through the saddle
point (the cross), the fissioning nucleus goes toward scission andZ0 and δ increase. At the same time, α decreases following thepotential slope as shown by the arrow 2 in Figure 6. As is seenfrom the sample trajectory in Figure 2, after passing through thesaddle point, the deformation δ increases from 0.2 to 0.4 withlarge fluctuation (around the number 3) while the increase of
Z0 is relatively small. With the increase of δ, the bump due tothe shell effect becomes prominent at Z0 = 0.75 and α = 0 in Figure 7. Since the Brownian particle keeps away from this bump,the mass asymmetry α increases again (the arrow 4) and thefission nucleus goes to asymmetric direction. This is how theasymmetry of fission fragments is determined.
It should be noted that the potential around Z0 = 1.2 ∼ 1.5 isvery flat in α direction in the present system. In addition, themass asymmetry α at the scission point differs from the one at
the saddle point. Therefore, it is inappropriate to determine themass asymmetry α by the potential valley; the dynamics after
the saddle point plays a very important role for the determination of the mass asymmetry at scission. The mass asymmetry
distribution cannot be estimated only from the position of thesaddle point such as the discussion with the static calculation.
Thus, the dynamical calculation is very important for the understanding of the fission modes.

stand fission paths. We solved the 3-dimensinal Langevin equation numerically in the potential energy including the shell correction and investigated fission modes for the system of 270 Sg.
From the calculation results, we clearly found two fission
modes. The one of these fission modes corresponds to the Standard II mode that has large deformation and has lower TKE. In
the mass distribution of the fission fragments of this mode, themass asymmetric fission dominates. This result is in qualitative
agreement with the experimental one. However, the peak energyof the TKE distribution of the fission fragments is larger than theexperimental result by about 30 MeV and this point is yet to beclarified.
We also investigated the influence of the potential energy on
the dynamics to understand the result of the mass asymmetric
distribution. We found that the mass asymmetry at the scission
is determined by the dynamics after the saddle rather than theposition of the saddle point. Therefore, we conclude that in order to understand fission modes, the dynamical calculation isvery important.
As a future study, we need the 4-dimensional Langevin calculation taking account of the independent deformation of eachfragment in order to include the deformed shell effect. It is also
necessary to include the change of the nuclear temperature dueto the particle emission. These studies are now under progress.

4. Summary
We studied the problem of fission mode by using the 3dimensional Langevin equation on the basis of the fluctuationdissipation dynamics and discussed the mass and the TKE distribution.
The static calculation in which the potential energy is calculated such as the works by Pashkevich is not sufficient to understand the mass-energy distribution. The dynamical calculationthat takes account of the shell effect is needed in order to under-Acknowledgement. We would like to thank Dr. Yamaji foruseful discussions and for supplying the code TWOCTR.
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