Journal of Nuclear and Radiochemical Sciences, Vol. 5, No. 1, pp. 1-5, 2004

ArticlesDynamical Calculation of Multi-Modal Nuclear Fission of Fermium Nuclei
T. Asano,*, a T. Wada,a M. Ohta,a T. Ichikawa,b S. Yamaji,c and H. NakaharadaDepartment of Physics, Konan University, 8-9-1 Okamoto, Kobe 658-8501, Japan
Japan Atomic Energy Research Institute, Tokai-Mura, Ibaraki 319-1195, Japanc
Cyclotron Center, RIKEN, Wako, Saitama 351-0198, Japand
Graduate School of Science, Tokyo Metropolitan University, Tokyo 192-0397, Japanb

Received: November 6, 2003; In Final Form: February 20, 2004
Multi-modal nuclear fission of fermium isotopes at low excitation energy is dynamically investigated. The multidimensional Langevin equation is used for the dynamical calculation. We use the potential energy taking accountof the microscopic energy which depends on the excitation energy. We calculate the mass and the total kinetic
energy (TKE) distributions of the fission fragments for 264Fm. The deformation of the fission fragments at thescission plays an important role in classifying the nuclear fission modes. We show that there exist following three
types for the scission configuration classified by the shape of the fission fragments: mass-symmetric compact
configuration, mass-asymmetric configuration, and mass-symmetric elongated configuration. We calculate theaverage mass number and the average TKE for each configuration. We also study the isotope dependence of themass distribution of the fission fragments from three fermium isotopes.

1. Introduction
In studying the multi-modal nuclear fission, fermium nuclei
are very important from the viewpoint of magicity of thedaughter nuclei since they decay into two proton-magic (Z = 50)
nuclei Sn in the mass-symmetric fission. In particular, thenucleus 264Fm decays into two identical double-magic (Z = 50,
N = 82) nuclei 132Sn.
It was found that there is a strong isotope dependence in the
fragment mass distribution of the spontaneous fission of fermium
nuclei; a mass-symmetric peak is observed for 258,259Fm, while
mass-asymmetric peaks are obtained for 254,256,257Fm.1−6 For thethermal neutron induced fission of 255 Fm, which should becompared with the spontaneous fission of 256 Fm, a masssymmetric peak is observed in the total mass distribution; thefragment mass distribution depends also on the excitationenergy. For the thermal neutron induced fission of 255Fm, theauthors of Reference 1 cut the total kinetic energy (TKE) distribution into several energy bins and studied the mass distribution for each bin. They obtained a sharp mass-symmetric
distribution for high TKE bins (TKE > 220 MeV) while theyobtained mass-asymmetric distribution for low TKE bins (TKE
< 220 MeV). Namely, the total mass distribution is composedof at least two components: a mass-symmetric component withhigher TKE and a mass-asymmetric component with lower TKE.
For the spontaneous fission of 258Fm, the authors of Reference 5
fitted the TKE distribution by two Gaussians and found that thehigh TKE component (TKE > 220 MeV) corresponds to a sharp
mass-symmetric distribution while the low TKE one (TKE <220 MeV) corresponds to a broad mass-symmetric distribution. In the both cases of the thermal neutron induced fissionof 255 Fm and the spontaneous fission of 258 Fm, the fission
events with the high TKE were found to form a narrow masssymmetric distribution.
The mass distribution and the TKE distribution consist of
several components and these components correspond todifferent nuclear fission modes. 1−11 Such a phenomenon isobserved not only for fermium nuclei but also for othernuclei.7−11 It is known that only the mass-symmetric fission
path exists in the potential energy surface (PES) given by theliquid drop model (LDM), so the shell effect plays a crucialrole for the multi-modal nuclear fission.
In order to understand the multi-modal nuclear fission,
many theoretical efforts have been done so far. One is themethod conjecturing the mass distribution from the PES andanother is dynamical calculation. The former has given an
explanation for the existence of several fission paths for264
Fm.12, 13 This approach has been applied to many nuclei, but
dynamical effects, such as the dissipation, are not included.
Dynamical calculations using the multi-dimensional Langevinequation have been applied at high excitation energies and they
explain the experimental data such as pre-scission neutron
multiplicity.14 Since the shell correction energy exists at low
excitation energies, e.g. at energies lower than 30 MeV, theinclusion of the shell correction energy is very important in the
dynamical calculation of the multi-modal nuclear fission. Thishas been pointed out for the fission of trans-actinide nuclei15
and for the fusion-fission mechanism of the superheavy
nuclei. 16 In this paper the dynamical calculation with thepotential energy including the shell effects is performed to
discuss the multi-modal nuclear fission of fermium isotopes.
Section 2 is a reminder of the multi-dimensional Langevinequation and the potential energy. Calculated results arepresented in sec. 3. Summary is given in sec. 4.
2. Methods
Dynamical approach based on the multi-dimensional
Langevin equation has been successfully applied to nuclear
reactions, i.e., fission and fusion-fission.14−18 In this study, weapply this approach to investigate the multi-modal nuclear
fission in the fermium nuclei. The time evolution of thefission process in the multi-dimensional deformation space istraced starting from the ground state of the fissioning nucleus
via saddle points to various scission configurations.
The multi-dimensional Langevin equation has the following
form;*Corresponding author. E-mail: dn121001@center.konan-u.ac.jp.
FAX: +81-78-439-2539.

© 2004 The Japan Society of Nuclear and Radiochemical Sciences
Published on Web 03/10/2004

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J. Nucl. Radiochem. Sci., Vol. 5, No. 1, 2004dqi
= (m−1)ij pj ,
dt
dpi
1 ∂
∂V
=– i –(m−1)jk pj pk – γ ij (m−1)jk pk + gij Rj (t),∂qi 2 ∂qidt

where qi denotes a coordinate in the deformation space and pi
is its conjugate momentum. The summation from 1 to n over
the repeated indices is assumed with n being the number ofcollective degrees of freedom. V(q) is the potential energy andmij (q) and γ ij (q) are the shape-dependent collective inertia anddissipation tensors, respectively. The hydrodynamical inertia
tensor is adopted with the Werner-Wheeler approximation for
the velocity field.19 The dissipation tensor is calculated withthe one-body type wall-and-window formula.20 The normalized random force Ri (t) is assumed to be a white noise, i.e.,〈Ri (t) = 0, 〈Ri (t1)Rj (t2) = 2δ ij δ (t1 – t2),〈〈

The strength of the random force gij is given by Σ k g ik g jk = Tγ ij,
where T is the temperature of the compound nucleus. It is
calculated from the excitation energy EX as EX = aT 2, where ais the level density parameter for a spherical nucleus. The
potential energy is calculated using the macroscopic-microscopic method. The macroscopic part of the energy is calculated with the Yukawa plus exponential model 21 and themicroscopic part is calculated with the Strutinsky shell correction method 22, 23 using the two-center harmonic oscillator
single-particle potential.24−27 The potential energy is expressed
as a sum of the two terms,

treat RC.M. (the distance between the centers of mass of future
fission fragments), δ (the deformation of the fission fragments), and A1 (the mass number of a fission fragment with A2
being the mass number of the other fission fragment) as thethree collective parameters. In order to obtain the deformationof the two fission fragments by a single parameter δ, we introduce the following relation for the ratio of the two axes a i andbi (see Figure 1):ai
2Ai 3 + δ=(i = 1, 2),bi
A 3–2δ
where A is the mass number of the fissioning nucleus. Figures
2 and 3 show various nuclear shapes that can be described withthe three collective parameters. Figure 2 shows the nuclear
shapes in the mass-symmetric fission and Figure 3 shows thenuclear shapes for the mass-asymmetric division with δ = 0.0.
In the present study, we used TWOCTR of two-center shellmodel code to calculate the potential energy surface. 26, 27, 30
The origin of the potential energy is set so that the macroscopic energy for the spherical shape vanishes. We do not takeaccount of the effect of the angular momentum or the particle
evaporation in this study.

V(q, EX) = EMacro(q) + EMicro(q, EX).
The macroscopic energy consists of the nuclear potential
energy ENuclear and the Coulomb energy ECoulomb.21 The microscopic energy is calculated as a sum of the shell correctionenergy ∆ E s and the pairing correlation correction energy∆ E pc . 28 The microscopic energy depends on the excitationenergy (temperature) since the occupation probability of thesingle particle levels depends on the excitation energy. The
excitation energy dependence of the microscopic energy isintroduced in the following form based on Ignatyuk’s suggestion;29
EMicro(q, EX) = EMicro(q, EX = 0)exp(–EX /Ed),

Figure 2. Various nuclear shapes in the deformation space for the
mass-symmetric fission of 264Fm. The abscissa denotes the distance
between the centers of mass of future fission fragments and the ordinate the deformation parameter δ . R is the radius of the parent
nucleus. The solid line shows the scission line.

where EX is the excitation energy and Ed is the shell damping
energy. We take Ed as 20.0 MeV in this study.
The nuclear shape is expressed by three parameters. We

a2 b2

b1

a1

Figure 1. Illustration of ai and bi that determine the fragment deformation.

Figure 3. Nuclear shapes in the deformation space with δ = 0.0 for264
Fm. The abscissa denotes the distance between the centers of massof future fission fragments and the ordinate the mass number of theheavy fission fragment. R is the radius of the parent nucleus. The
solid line shows the scission line.

Dynamical Calculation of Multi-Modal Nuclear Fission

3. Results
First, we investigate the fission of 264Fm. Figures 4 and 5
show the mass distribution and the TKE distribution of thefission fragments at the excitation energy EX =10.0 MeV. One
observes a single peak for the mass distribution and threepeaks for the TKE distribution. Since the fission fragments inthe mass-symmetric fission of 264Fm are double-magic nuclei,
it is expected that the scission configuration is compact. The
TKE of the fission fragments essentially depends on the
distance between the centers of mass of fission fragments.
Thus the TKE of the fission fragments depends sensitively on
the compactness of the scission configuration. The masssymmetric compact scission configuration corresponds to thehighest TKE. It is expected that the other two TKE peaks
correspond to more elongated scission configurations.
As is shown in Figure 2, the compactness of the scission
configuration is governed by the fragment deformation parameter δ. Therefore, we pay special attention to the deformation parameter δ at the scission configuration. In Figure 6, weshow the distribution of the deformation parameter δ at thescission configuration. From the figure, we observe that this
distribution consists of three components similarly to the TKE
distribution shown in Figure 5. It is expected that there is acorrelation between the structures appearing in Figures 5 and6. We classify the fission events into three components
according to the values of the deformation parameter δ at thescission configuration; Component I corresponds to δ < 0.04,
Component II corresponds to 0.04 < δ < 0.26 and Component
III corresponds to δ > 0.26. In Figure 7, the TKE distribution

Figure 4. Distribution of the mass number of the fission fragments
for the fission of 264Fm at the excitation energy EX = 10.0 MeV.

Figure 5. Distribution of the TKE of the fission fragments for thefission of 264Fm at the excitation energy EX = 10.0 MeV.

J. Nucl. Radiochem. Sci., Vol. 5, No. 1, 2004

3

shown in Figure 5 is decomposed into these three components.
It is seen that Component I corresponds to the events thatbelong to the highest TKE peak, Component III to the lowest
TKE peak, and Component II lies in-between. As for the TKE
distribution, we can thus understand that the structure seen in
Figure 5 corresponds to the classification in the δ-space.
The next problem is the mass distribution of each component classified by the value of δ at the scission. In Figure 8,

Figure 6. Distribution of the deformation parameter δ at the scission
for the fission of 264Fm at the excitation energy EX = 10.0 MeV.

Figure 7. Decomposition of the TKE distribution according to thethree components of the deformation parameter δ . Open circles
denote Component I, solid upward triangles Component II, and open
downward triangles Component III. Solid squares denote the sum of
the three components.

Figure 8. Decomposition of the mass distribution according to thethree components of the deformation parameter δ. The meaning ofeach symbol is the same as that in Figure 7.

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J. Nucl. Radiochem. Sci., Vol. 5, No. 1, 2004

the total mass distribution is decomposed into the three components. It is found that Component I and Component III correspond to the mass-symmetric fission and Component II to the
mass-asymmetric fission.
In order to study the origin of two components for the masssymmetric peak and one mass-asymmetric peak, we investigate
the PES. First, Figure 9 shows the PES for the mass-symmetricfission of 264Fm. One can see that the ground state locates at
RC.M./2R ~ 0.45, δ ~ 0.10 (V ~ −0.70 MeV). It is found that
there are two fission paths; one leads to Component I and theother to Component III. The path for Component I has only
one saddle point (RC.M./2R ~ 0.50, δ ~ 0.25, V ~ 3.00 MeV),
while that for Component III has the second minimum(RC.M./2R ~ 0.62, δ ~ 0.30, V ~ −1.00 MeV) and the second
saddle point (R C.M. /2R ~ 0.70, δ ~ 0.30, V ~ −0.10 MeV).
Therefore, the yield of Component I is expected to be greaterthan that of Component III.
Next, Figure 10 shows the PES in the case of δ = 0.135
which corresponds to the deformation peak of Component II.
We find the mass-asymmetric second saddle point (RC.M./2R ~0.70, AH ~ 147), while we find only a mass-symmetric fission
valley after the second saddle. It is difficult to predict where
the peak of the mass distribution will locate only from thisPES. Thus we need the dynamical calculation. In comparison,
Figure 11 shows the PES in the case of δ = −0.11 which corresponds to the deformation peak of Component I. In this case,one finds the mass-symmetric saddle point and the masssymmetric fission valley.
Zhao et al. investigated the systematics of the TKE in theactinide region.10, 11 They assumed that the TKE is equal to the
Coulomb energy at the scission configuration for point
charges, namely TKE = Z1Z2e2/D, and defined the shape elongation at the scission point by β = D/(R1+R2), where R1,2 are
the radii of fission fragments. They showed that there aremainly two types of scission configuration: a mass-asymmetric(AH = 140) configuration with β = 1.53 and a mass-symmetricelongated configuration with β = 1.65. Additionally, there is amass-symmetric compact configuration with β = 1.33 in theheavy region where the mass number of the fissioning nucleus
is nearly equal to 260. In order to compare our results withthis experimental systematics, we calculate the peak values andthe widths of the mass number and TKE distributions for eachcomponent from the Gaussian fitting. The results are listed inTable 1. One finds that the shape elongation β of Component I
corresponds to the mass-symmetric compact configuration,that of Component II to the mass-asymmetric (A H = 140)
configuration and that of Component III to the masssymmetric elongated configuration. However, the mass
number of the heavier fragment in the mass-asymmetry modeis predicted to be 147 and is different from the experimental
value A H = 140. Furthermore, our shape elongation β isslightly bigger than that of the experimental systematics for the
mass-symmetric elongated configuration.
The fermium nucleus 264Fm has not been studied experimentally, but other fermium isotopes 256,258Fm have been studied.
A remarkable isotope dependence is observed experimentally;in the spontaneous fission, the mass-symmetric peak isobserved for 2 5 8 Fm, while the mass-asymmetric peak is
TABLE 1: Peak Values and Widths of the Mass and TKE
Distributions Obtained from the Gaussian Fitting to Each
Component<A1>

Figure 9. The contour plot of the PES for the mass-symmetric fissionof 264Fm. The abscissa denotes the distance between the centers ofmass of future fission fragments and the ordinate the deformation parameter δ. R is the radius of the parent nucleus. The first and thesecond saddle points are marked by X’s. Arrows show expectedfission paths.

Figure 10. The contour plot of the PES in the case of δ = 0.135 for264
Fm. The abscissa denotes the distance between the centers of massof future fission fragments and the ordinate the mass number of afission fragment. The solid line shows the mass-symmetric division.
R is the radius of the fissioning nucleus. Arrows show expected fission
paths.

σ (A1)

<TKE>
/ MeV

σ (TKE)/ MeV

βComponent I

131.5

6.80

232.1

6.30

1.31

Component II
Component III

147.0
132.3

9.10
34.2

200.8
171.7

10.4
15.0

1.50
1.77

<A 1 >: the peak mass number, σ(A 1 ): the variance of the mass
number, <TKE>: the peak TKE, σ(TKE): the variance of the
TKE, β : the shape elongation deduced from <A1> and <TKE>.

Figure 11. The same as Figure 10, but for the case of δ = −0.11.

Dynamical Calculation of Multi-Modal Nuclear Fission

Figure 12. Isotope dependence of the mass distribution for the fissionof 256,258,264Fm at the excitation energy EX = 10.0 MeV. The solid
squares denote the mass distribution of the fission fragments of 256Fm,
open circles that of 258Fm, and open triangles that of 264Fm.

observed for 256Fm.1−6 We calculate the mass distributions for
256,258
Fm in the same manner as 264Fm. We compare the mass
distributions of 256,258,264Fm in Figure 12. When we compare
the full width at half maximum (FWHM) for each mass distribution, this value increases from 264Fm (FWHM = 25) to 258Fm(FWHM = 30) and then further to 256Fm (FWHM = 33). We
did not find the drastic isotope dependence that was found in
the spontaneous fission. It should be noted that our calculation
has been performed for the excitation energy EX = 10.0 MeV
and that mass-symmetric peaks have been observed in thethermal neutron induced fission of both 255Fm and 257Fm.
4. Summary
Multi-modal nuclear fission of the fermium nuclei 256,258,264Fm
at the low excitation energy was dynamically investigated. We
payed special attention to the deformation parameter δ which
essentially controls the scission configuration. We classified
the fission events in three components according to the valuesof the deformation parameter δ at the scission configuration:
Component I (δ < 0.04), Component II (0.04 < δ < 0.26) andComponent III (δ > 0.26). The mass and TKE distributions arewell classified by the δ decomposition. It was found that theshape elongation β which is deduced from the peak mass
number and the peak TKE for each component is consistent
with Zhao’s experimental systematics. As a conclusion, weemphasize the importance of the deformation parameter δ atthe scission configuration in the dynamical calculation. We
are planning to extend this calculation to the four-dimensional
one, where the deformation of the two fission fragments isexpressed not by a single parameter δ but by two independent
parameters δ1 and δ2.
Acknowledgements. The authors wish to express their
gratitude to Dr. Y. Nagame and Dr. Y. L. Zhao for their useful
discussions.
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