Calculation of Fission Yield by Macroscopic-Microscopic Method
Based on Selective Channel Scission Model

Masayuki OHTA and Shoji NAKAMURA
Japan Atomic Energy Agency
2-4 Shirakata-Shirane, Tokai-mura, Naka-gun, Ibaraki, 319-1195, Japan
E-mail: ohta.masayuki@jaea.go.jp

The mass-distributions of fission yields for neutron-induced fissions of U-235 were calculated bya macroscopic-microscopic method based on the selective channel scission model. The present result wascompared with the previous result from the aspect of fission modes.

1. Introduction
The selective channel scission (SCS) model has been proposed and developed to calculate fission
yields for any nuclei [1-3]. The SCS model deals with the fission process for each channel. The fission
yield is obtained from the penetrability of the “channel-dependent” fission barrier. In previous analysis [3],mass-distributions of fission yields were calculated on simple assumptions about the channel-dependent
fission barriers. This calculation method is applicable to wide range of fissionable nuclei without adjustableparameters. However, there were discrepancies between the calculated results and experimental data offission yield in the mass regions of A = 85–95 and A = 140–150.
In this work, the channel-dependent fission potentials were calculated by a macroscopicmicroscopic method based on the idea of SCS. The mass-distribution of fission yield was calculated for theneutron-induced fission of U-235.

2. SCS Model and Calculation of Fission Potential
The SCS model deals with the fission process for each channel. The fission yields are calculated
from the penetrabilities of the “channel-dependent” fission barriers Ef.
The basic definition of nuclear shape is given byN⎛⎞
R(θ ) = λ−1 R0 ⎜1 + ∑ α n Pn (cosθ )⎟ ,⎝ n =1⎠(1)where λ-1 is the volume conservation, R0 is the radius of spherical nucleus, αn is the deformation parameterand Pn is Legendre polynomial.
A macroscopic-microscopic method is commonly used for the calculation of fission potential.

2006 Symposium on Nuclear Data

SND2006-V.06-1

The total potential energy E of a deformed nucleus is defined as the summation of the liquid-drop energy
ELDM as a macroscopic term and the shell correction energy Eshell as a microscopic term in this method.

E = E LDM + Eshell .

(2)

The ELDM is derived from the surface energy ES and the Coulomb energy EC of the deformednucleus.

ELDM = ES + EC .

(3)

The channel-dependent fission potentials were calculated by a macroscopic-microscopic method
based on the idea of SCS. The surface energy ES in the macroscopic term was obtained from an equation
whose form was proportional to the surface area S of the deformed nucleus [4].

ES = γ S ,γ=2⎡
1⎛ N −Z ⎞ ⎤1
κ−a⎟⎜⎥.
2⎢
4π r02 ⎢⎣⎝ A ⎠ ⎥⎦(4)

The Coulomb energy EC also in the macroscopic term was obtained by the Monte-Carlo integralof the Coulomb energy between differential volumes which were taken at random all over the region of thedeformed nucleus (see Fig.1). Mersenne Twister [5] was used as a random number generator.

EC =1 ⌠⌠ 1 ρ i ρ j 3 3d ri d rj⎮
2⎮⌡⌡ 4π ε 0 rij(5)ρρZe1=
ΣΣ i 2 j 1 .
2 i≠ jρ rij
4π ε 0 2N2NThe shell energy Eshell in the microscopic term was calculated approximately as follows. The twofission fragments (FP1 and FP2) were assigned to the shape of the deformed nucleus for a channel (see Fig.
2). The whole shell energy was calculated from the sum of the shell energies of the two deformed fission
fragments assigned to the deformed nucleus (Esh1 and Esh2).

Eshell = Esh1 + Esh2 .

(6)

A calculation code [6] was used for the calculation of shell energy for each fission fragment.
FP1FP2

ρ1(Z1,A1)(Z2,A2)3

r23

1r122Rρ2

r34

FP1FP24Fig. 1 Differential volumes taken in theFig. 2 Assignment of the two fragments todeformed nucleus

the deformed nucleus

2006 Symposium on Nuclear Data

SND2006-V.06-2

The potential near the saddle point is approximated by the inverted parabola and the curvatures αis assumed as a constant for all humps, for simplicity. The tunnel probability Pi for the saddle point i isreduced asPi ≈1,
1 + exp[0.218 α μ ΔEi ](7)

in MeV and fm units, where μ = A1A2/(A1+A2), ΔEi = Efi - Ex and A1 and A2 are the mass number of FP1and FP2, respectively. In case of a two-humped potential, the probability P is deduced from tunnel
probabilities for the two humps (PA and PB).

P=PA PB.
PA + PB(8)

The fission yields are obtained by summing up these probabilities all over fission channels.

3. Results and Discussions
Figure 3-(a) shows an example of calculated fission potential for a channel. These potential
calculations were carried out for about 230 channels that have high fission yields. The tunnel probability P
was obtained for an excitation energy (e.g. Ex = 0), as shown in Fig. 3-(b). The parameters α2 at inner andouter saddle points were shown in Figs. 3-(c) and 3-(d), respectively.
Fission yields for the thermal neutron-induced fissions of U-235 were obtained as shown in Fig.
3-(e). The α was taken as 0.2 in Eq. (7). Prompt neutron emission was not considered in the calculated
fission yield. Meanwhile, the prompt neutron emission is considered for JENDL-3.3 data. It is known thatthe neutron multiplicity against mass number of fragment shows a saw-tooth curve [7]. Then, the calculated
fission yield showed qualitative consistency with the data of JENDL-3.3. There were discrepancies in mass
regions of A = 85–95 and A = 140–150 in previous analysis [3]. Although there were not such discrepancies
in present result, fission yields were underestimated in mass regions above A = 150 and below A = 90.
The shape elongation factor η was obtained at the saddle point deduced from JENDL-3.3 data ina previous analysis [2]. The η changed the trend at mass of fragments A ~ 130. The similar behavior
appeared in the deformation parameter a2 in Fig. 3-(c). The α2 contributes significantly to the deformationof nucleus. It might depend on the existence of symmetric and asymmetric fissions.

4. Conclusions
The channel-dependent fission potentials were calculated for the neutron-induced fission of U-235
by a macroscopic-microscopic method based on the selective channel scission model. The massdistribution of fission yield was obtained for thermal neutron-induced fission of U-235.

2006 Symposium on Nuclear Data

SND2006-V.06-3

A
60

3080

100120140160180

1.E+00

1.E-052015PE [MeV]25

1.E-1010
1.E-15500

0.32 0.64 0.96 1.28

1.6

1.E-201.92 2.24 2.56 2.88

α2U-236v1.1 5.47+5.46 Ex=0(b) Example of tunneling probability P(a) Example of fission potential

0.8

2.5

0.7

20.6
1.5

α2

α2

0.5
0.4

10.3
0.5
0.2
0.1118128138148

158

A

168

0118

128

138(c) Parameter α2 at inner saddle point

100148158

AU-236 5.47v1

168
U-236 5.46(d) Parameter α2 at outer saddle point

U-235

JENDL3.3 (Thermal Neutron Fission)SCS (5.47+5.46 v1.1)

FP YIELD (%)

1010.10.010.001
6080100120140
MASS NUMBER

160180(e) Fission yield for thermal neutron
Fig. 3 Results for n+235U

2006 Symposium on Nuclear Data

SND2006-V.06-4

Acknowledgement
The computations were carried out with Fujitsu PRIMEPOWER HPC2500 at Japan Atomic Energy
Agency (JAEA).

References[1] A. Takahashi et al., Jpn. J. Appl. Phys., 40 (2001) 7031.
[2] M. Ohta et al., Jpn. J. Appl. Phys., 45 (2006) 6431.
[3] M. Ohta et al., Proc. 2005 Symp. Nucl. Data, JAEA-Conf 2006-009 (2006) p. 145.
[4] W.D. Myers and W.J. Swiatecki, Nucl. Phys., 81 (1966) 1.
[5] M. Matsumoto and T. Nishimura, ACM Trans. on Modeling and Computer Simulation, 8 (1998) 3.
[6] F. Garcia et al., Comp. Phys. Comm., 120 (1999) 57.
[7] For example, The New Encyclopaedia Britannica, Encyclopaedia Britannica, Chicago, 1980 15th ed.,Vol. 13, p. 301.

2006 Symposium on Nuclear Data

SND2006-V.06-5