Journal of Nuclear and Radiochemical Sciences,Vol. 3, No. 1, pp. 103–107, 2002

103

Fission Characteristics of Individual Deformation Paths in Heavy Elements
Y. L. Zhao,∗,a,b I. Nishinaka,c Y. Nagame,c K. Tsukada,c K. Sueki,b M. Tanikawa,d S. Goto,e and
H. NakaharabaInstitute of High Energy Physics, Chinese Academy of Science, Beijing 100039, ChinabTokyo Metropolitan University, Tokyo 192-0397, Japanc
Japan Atomic Energy Research Institute, Tokai, Ibaraki 319-1195, Japand
Department of Chemistry, The University of Tokyo, Tokyo 113, Japane
Department of Chemistry, Niigata University, Niigata, 950-2181, Japan
Received: November 13, 2001; In Final Form: April 3, 2002
The deformation properties of heavy nuclei undergoing the individual fission paths are studied. To understand thenuclear mass division process at low energy, fission events for the symmetric and asymmetric fission paths areanalyzed from the overall fission events. The fragment mass-yield distributions for the symmetric and asymmetricfission paths in the fission processes of 210 Po, 227 Ac, 233 Pa, 249 Bk, and 259 Md are systematically studied. Fission
characteristics including the deformation of the fissioning nucleus at the scission configuration, the fragment massyield distributions, and the total kinetic energy release for individual fission paths are presented.

1. Introduction
Theoretically, multiple fission paths were recently found by
the theoretical calculations of the fission potential surface in afive-dimensional deformation space at and beyond the outer saddle points.1 Experimentally, by examining excitation functions
for fission events in the fragment mass distribution and the total
kinetic energy (TKE) distribution, we demonstrated the presence of correlations between the saddle-point and scission-point
configurations, i.e. the existence of the independent fission pathsin the dynamic descent of deforming nuclei from the saddle toscission configurations.2 In the lighter actinides, 232 Th and 238 U,
the existence of two independent fission paths has been experimentally certified.2 One initiating with the lower threshold energy reaches the less elongated scission configuration leading to
the asymmetric mass division and the higher fragment kinetic
energy, and the other path starting with the higher threshold energy reaches the elongated scission configuration leading to thesymmetric mass division and the lower fragment kinetic energy.
The shape of the mass-yield distribution is one of the longstanding puzzles in fission observations. The observed fragment
mass-yield curve was analyzed as a superposition of events originating from different types of the fission process.3–7 It has been
turned out that these fundamental processes essentially represent the independent fission paths,8 or the deformation paths9–11
in fission. A recent experiment studying the Coulomb fission ofmass-separated radioactive beams unveiled a systematical variation of the shapes of the mass-yield curves from the single
symmetric peak to the triple-humped peak and to the doublehumped asymmetric peak, for the isotopes of thorium with thechange of the neutron number from 132 to 139.12 In fact, nuclear fission essentially represents an elongation process including many kinds of collective motions of an atomic nucleus. Nuclear mass division occurs at the end (scission configuration) tothe dynamic deformation motion of the fissioning nucleus. As
mentioned before, the dynamic motions of nuclear matters in
the multidimensional space from saddle toward scission points
proceed along at least two different paths. The properties offission fragments are determined by the characteristics of thesedeformation paths. Accordingly, the experimental informationsof each deformation paths of the fissioning nuclei are capableof providing important clues for better understanding the fission
properties, such as the mass-yield distribution observed in theexperiment. In this context, our present attempt was made to∗Corresponding author. E-mail: zhaoyuliang@ihep.ac.cn.
FAX: +86-(010)-8821-2853.

understand the fission observations from the viewpoint of thecharacteristics of the individual fission paths.
2. The Deformation Properties of Individual Fission Paths
The measured average TKE(A1 , A2 ) value released from agiven mass split process is, in first order approximation, equal to
the Coulomb repulsion energy between two nascent fragments
A1 and A2 , i.e. TKE(A1 , A2 ) = Z1 Z2 e2 /D(A1 , A2 ). Z1 and Z2 are
the charges for corresponding fragment masses A1 and A2 . They
were obtained from the UCD model by which the ambiguity being brought into the results was turned out to be about 1%.10 The
D(A1 , A2 ) (fm) term is the distance between the charge centersof a pair of complementary fragments at the scission configuration which is the last state for the fissioning nucleus still retaining as a whole. In order to allow an estimation of the degreeof deformation at the scission point for different nuclei, a shapeelongation β, was defined as β = D(A1 , A2 )/D0 (A1 , A2 ). Where,
1/3
D0 (A1 , A2 ) = r0 (A1/3
1 + A2 ) with r0 = 1.17, means the distanceof two charge centers for the spheres. The β is hence a measureof the degree of the deformation of the scissioning nucleus andan indicator of how much, prior to fission, the nucleus deviatesfrom its spherical shape.
In Figure 1, the measured shape elongations of the scissioning nucleus leading to individual mass splits are displayed asa function of the mass ratio (AH /AL ) of the pair fragments, inpart (a) for fissioning systems 3 He + 232 Th with EHe = 60 MeV,in part (b) for p + 232 Th with E p = 14.7 MeV, and in part (c) forp + 238 U with E p = 14.7 MeV by solid circles. The data begin
from the symmetric center at A1 = A2 = A f /2. The plotted data
include the fission results of nuclei in the light actinide region,
233
Pa, 235 U, and 239 Np at the energies ranging from ∼15 MeV to
60 MeV. On the results of the mass division, fission of 60 MeV
3He + 232 Th produced a mass-yield distribution having a broad
symmetric shape13 that represents the typical feature of the symmetric fission process. While fission of p + 232 Th and p + 238 U
at E p = 14.7 MeV produced the mass-yield distributions having
double-humped peaks that represent the mixture results of coexistence of the both processes of the symmetric and asymmetricmass division. Thus, to understand the nuclear mass division
process occurring at the low energy, we need to distinguish thefission events for the symmetric and asymmetric processes fromthe overall fission events observed.
From the experimental data of solid circles, one sees thatin the symmetric mass division region, the observed β of thescissioning nuclei (solid circles around the symmetric mass di-

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

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

Zhao

vision with AH /AL ∼ 1) show a nearly constant value. In theasymmetric mass division region (solid circles in the area of
AH /AL > 1.5), the observed β values of the scissioning nuclei
in parts (b) and (c) also remain a nearly constant, but the absolute value is different from that observed in the symmetric mass
division region. The question rising here is the β values in between the above two regions are largely varying with the AH /AL .
They no longer remain the constant nature in between the symmetric and the asymmetric regions of mass divisions. The interpretation for this phenomenon is that fission events in this mass
region are produced by more than one type of the fission processes. Under the assumption of the symmetric and asymmetricfission, we decomposed the overall fission events into two components, for the details of the analysis method the reader may
refer to Reference 14. It is noted here that this assumption of
two fission components does not deny the existence of more fission paths. Two were considered here because the residuals of
the fission events after two-component analysis (for the studied
fissioning systems) were less than 5% of the total fission. We
also tested that adding more fission paths into the analysis did
not essentially alter the results presented in this paper.
Open circles and open triangles are the results for the symmetric and asymmetric fission paths, obtained from the analysisof the TKE(A1 , A2 ) events using the same analytical method in
Reference 14. The data of open circles were determined by the
initial parameters for the symmetric fission paths, according to
the newly discovered regularity: invariance of the final deformation of fissioning nuclei.9 The results of open circles were obtained from the analysis, showing the β values for the asymmetric mass division in the mixture region. One can find that the analytical data (open triangles) in the region of 1.1< AH /AL < 1.5
show almost identical value with the observed data (solid circles) in the region of AH /AL > 1.5. This is to say that nuclei
undergoing the asymmetric fission path reach a nearly identicalscission configuration prior to rupturing into fragments, because
the β value is almost unchanged with the change of the mass

2
1.8
1.6
1.4
1.2
1
1.8
1.6
1.4
1.2
1

1.8
1.6
1.4
1.2
1
1

split.
Two remarks need to be noted here. The first, the β values
for the nuclei undergoing both the symmetric fission path (seepart (a) of Figure 1) and the asymmetric fission path (see parts(b) and (c) of Figure 1) show a slight dependence on the mass
split. The β variation of either fission path with the AH /AL (inthe region of AH /AL = 1.0–2.0) corresponds to the fluctuation of
the TKE ∼ 1.5 MeV. This magnitude, may reflect some important information of the fission processes, but unfortunately is
within the order of the experimental ambiguity. Therefore, only
from the present experimental data, it is difficult to clarify if
this variation carries some specific physical significance for themass division process of the nuclei. The second, the minimum β
value in the asymmetric fission path is observed at the mass split
leading to the fragment A = 132 (corresponding to AH /AL = 1.3
for p + 232 Th and 1.25 for p + 238 U reactions). This may attributeto the presence of the spherical shells N = 82 and Z = 50 in the
fragments, and hence provides the direct evidence for the effects of spherical shells on the final mass division process. In
Figure 2(a), the shape elongations for the nuclei undergoing thesymmetric fission paths are plotted versus the mass number ofthe fissioning nucleus. A part of data was reported in References 9, 10, and some new experimental data were added into
the present plots. One data point is the β value for one fissioning
nucleus as indicated by its name. The data in the area coveredby all dot-lines in Figure 2(a) are the β for the fissioning systems with the excitation energy ranging from 10 up to 120 MeV.
Skipping such a wide energy region, the fission characteristics
show a large variation, but one can find that the β value retains
a constant and the variation is as small as less than the verticalheight of the shadow space. On the other hand, the β values
along the horizontal direction also show a constant nature. It is
2
210

Po

208

205

212

Po

Bi

243

Am

239

225

Po

Np

233

Pa

U

258

245

Bk

240

Rf 260
Rf

256

Rf

Am

270

Sg

244

Cm

1.8

246

Fm

1.6

235

3

232

U ( He+

(a)

Th)

1.4

Bi

Bi

213

At

262

Rf

Pa

Ac

Th

259

233

227

222

Fm Lr

Es

Np

209

257

255

238
207

242

Pu

(a)

246

241

Cm

Am

239

Am

249

256

Bk

Am

βvalue for thesymmetric fission path

1.2

3

Cf

245

259
252

248

Fm

258

No

Fm

Fm

E( He)=60 MeV

260

Md 258No

259

Fm

262

No

259

Md

1
200

(b)
2

232

14.7 MeV p+

210

220

230

240

250

260

270

Mass number of the fissioning nucleus

β value forasymmetricfission path

1.8

Th

238

Np

249

Am

No

252

Pu

Np

256

Bk

245

239

234
233

Cf

259

Lr

Pa

246

241
230

Np

253

Fm

Am

236

Th

Es

244

228

Cm

240

Ac

I

Md

Cf

248

Pu242Pu

260

254

Fm

258

Rf

1.6

(c)
1.4

242

Pu
246

U

237

Pu

229

Th

1.2

Cm250Cf

234

(b)
Ac

259

Cm

U

239

231

Np

Pa

14.7 MeV p+

Fm

Bk
252

Pu

1
200

1.2

1.4

1.6
A /A
H

1.8

256

Cf

238

U

I

Md

256

245

No

238

I

248

236
227

No

Fm

Pu

213

At

258

254

240

2

L

Figure 1. The shape elongations of fissioning nuclei undergoing thesymmetric (open circles) and asymmetric (open triangles) fission paths
are plotted as a function of the mass ratio of a pair of complementaryfragments. The measured data points are indicated via solid circles.

210

220

230

240

250

260

270

Mass number of the fissioning nucleus
Figure 2. (a) The shape elongations for nuclei at the scission configuration in the symmetric fission path are plotted as a function of themass number of the fissioning nucleus. (b) The shape elongations fornuclei at the scission configuration in the asymmetric fission path areplotted as a function of the mass number of the fissioning nucleus. The
dot-lines marked areas in (a) and (b) indicate the vertical variations of
the values.

Fission Characteristics of Individual Deformation Paths

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

to say that the final shape elongations of nuclei undergoing thesymmetric fission path are nearly the same and not varied withthe mass number of the fissioning nucleus, giving the β ∼ 1.65.
The data indicated by solid and open squares are the β values for
spontaneous fission nuclei in the heavy actinide region. They
give different β value from those for the lighter actinide nuclei.
This difference in the final deformation of the heavy nucleus(A f ∼ 260) and the light nucleus (A f < 245) indicates that thesymmetric fission path in the light and heavy actinide is not thesame one. From the shape of the fragment mass-yield distribution, it is usually considered that the symmetric fission pathnearly vanishes in the region of A f ∼ 245, and a sudden changeof the fission mode occurs when the nucleus becomes heavier
than A f = 256. But the results in Figure 2(a) indicate a smoothtransition of the β from A f < 245 with the β ∼ 1.65 to A f ∼ 260with the β ∼ 1.33. No sudden change taking place in the fission
processes of nuclei is observed when the nucleus changing fromthe lighter to heavy actinides.
The data in Figure 2(b) are similar with those in 2(a) but for
the asymmetric fission path. Due to some discussions of theseresults have been made in Reference 9, we hence only mention
here some natures that are important for the argument in followings. One may have noticed the identical β values for theasymmetric fission path. All nuclei reach a degree of deformation with a constant β ∼ 1.53 at the scission configuration prior
to rupturing into fragments. In conclusion, in the asymmetricfission path the final degrees of the deformation are invariant
with the nuclear mass and charge of the fissioning system. The
Experimental data

300

New formula fromβ sym
Viola et al.
1

250

2

One-body dissipation0
3

Two-body dissipation

3

Surface-plus-Window0

No dissipation

space marked using dot-lines in Figure 2(b) indicates the verticalvariation for the β value in the studied A f -region, it is generally
less than 2% of the absolute value.
3. The Total Kinetic Energy Release in Individual Fission
Paths
From the identical shape elongation of fissioning nuclei, theformulas for the TKE release in the symmetric and asymmetric fission paths were derived. In the symmetric fission paththe constant β of 1.65 gives TKE = 0.1173(Z 2f /A1/3f ) + 7 MeV,
and in the asymmetric fission path the constant β of 1.53
14
gives TKE = 0.1217(Z 2f /A1/3
A comparison of thef ) + 4 MeV.
presently derived TKE function with the available experimental data is given in Figure 3. The experimental data are indicated using solid circles, including the results for the lowenergy symmetric fission path and the heavy-ion induced symmetric fission path. The experimental data are taken from References 15–17 and references therein. The new formula (derived
from the deformation degree) of the symmetric fission path isshown by the bold line, it is in excellent agreement with theexperimental data of solid circles. For comparison, the resultsof the newest version of Viola’s empirical TKE systematics,
0.1187(Z 2f /A1/3f )+7.3 MeV (Ref. 18) is also indicated by a solidline. It is surprisingly close to the newly derived one for thesymmetric fission path. Thus, we can now make a conclusionthat the simple linear dependence of the average TKE release infission on the coulomb parameter Z 2f /A1/3
found by Viola et al.
fessentially originates from the constancy of the shape elongationof the fissioning nucleus at the scission configuration.
In 1980’s, Nix and Sierk et al. developed a dynamical modelfor the nuclear fission process, the dashed lines with the num10

1

10

0

(A)

10

-1

1

10

-2

10
10

-2

0

10

-1

10

-2

227

ASym.

10

-1

10

-2

Ac

226

Ra)

Symm.

(C)

ASym.

233

Pa

232

(p+

Th)

Symm.

(D)

ASym.

249

0

Bk

248

(p+

Cm)

Symm.

(E)

150

0

Bi)

(p+

-1

10

50

209

0

10

100

Po

(p+

(B)

10

150

210

Symm.

2

200

105

ASym.

259

Md (SF)

100

0

500

1000

1500

2000

Z 2/A 1/3
f

f

Figure 3. The average total kinetic energies obtained from the experimental measurements (solid circles), the presently derived TKE
formula (bold line), Viola’s TKE systematics, and theoretical calculations of the dynamic model taking into account one-body dissipation(dashed line with sign “1”), two-body dissipation (dashed line with sign“2”), surface-plus-window dissipation (dashed line with sign “3”), andwithout dissipation (dashed line with sign “0”) are plotted versus the
coulomb parameter.

50

Symm.

0
60

80

100

120

140

160

180

Fragment mass number
Figure 4. The fragment mass-yield distributions for the representative
fissioning nuclei over a large region. They show the smooth, systematic changes in the grass characteristics (such as the shape and peak
position) of the mass-yield curve for either of the symmetric and asymmetric fission paths for nuclei from pre-actinide to the heavy actinide.

106 J.Nucl.Radiochem.Sci.,Vol. 3, No. 1, 2002
ber show their theoretical results.19 In the calculations, effects
of several types of the nuclear dissipation force on the dynamical evolution of the fissioning nucleus beyond the fission saddle
point were taken into account. The long-dashed line with thesign “1” is the result for the one-body dissipation, that with thesign “2” for the two-body viscosity, that with the sign “3” for thesurface-plus-window dissipation. The results calculated without
considering any dissipation force are shown via the short-dashedlines with the sign “0”. The theoretical prediction of the dynamical model with the surface-plus-window dissipation shows
the best agreement with the experimental data and data obtained
from the presently derived TKE formula, except for those in thefission of very heavy nuclei in the region of Z 2f /A1/3
f > 1800.
4. The Fragment Mass-yield Distributions for Individual
Fission Paths
On the mass-yield curve, the changing tendency of its shapewith the change of the fissioning nucleus in the heavy actinide
region was studied in Reference 20. Here we expand our viewto a wider region of the fissioning nucleus. In Figure 4, thesystematic changes of the mass-yield curve in the fission process of the preactinide ((A) 210 Po and (B) 227 Ac), and actinide((C) 233 Pa, (D) 249 Bk, and (E) 259 Md) are displayed. The data inparts (C) and (D) for the 233 Pa and 249 Bk are from present measurements. Those in parts (A), (B), and (E) for 210 Po (Ref. 21),227
Ac (Ref. 22), and 259 Md (Ref. 23) are taken from literature.
The corresponding fissioning nuclei and fissioning reactions areindicated in the upper corner of each plot. The solid lines show
the mass-yield curves for the symmetric fission path. For 210 Po,
227
Ac, and 259 Md fission, they were obtained from the measurements by experiments. For 233 Pa and 249 Bk, they were gained
from the TKE intensities by the analysis method described in
Reference 14.
In the following discussions, we will concentrate on the varying trend of the mass-yield curves in the symmetric and asymmetric fission paths. The peak positions of mass-yield curvesin the symmetric fission path are marked using a vertical arrow
with the letter “Symm”. The peak positions for the light fragment mass in the asymmetric fission path are marked using ahorizontal arrow with the letter “Asym”, and those for the heavy
fragment mass are marked by a dotted line in the vertical direction.
One sees that for preactinide in 210 Po region, the nuclei fission symmetrically and produce a broad single peak of the fragment mass-yield curve as the one shown in Figure 4(A), In 227 Ac
region, the fission process produces a triple-humped fragment
mass-yield curve as the one given in Figure 4(B), where thesymmetric and asymmetric fission paths are of the similar importance in determining the results of the nuclear mass division.
But as compared to those in the 210 Po region, the width of thesymmetric mass-yield curve becomes narrow. As going into theactinide region, fission of 233 Pa as well as 249 Bk gives out the
double-humped fragment mass-yield curves. This is a typical
pattern in the actinides. The results of bold lines from the 210 Po
to 249 Bk show that the width of the mass-yield curve for the symmetric fission path becomes small with the increase of the massand charge of the fissioning system. This is probably due to thecompetition arising from the occurrence of the asymmetric fission path of the dotted lines in Figure 4(B). From the viewpointof the fission barriers (threshold energies), it has been found thatthe fission barrier-heights for the symmetric and asymmetric fission paths have the comparable values in the 227 Ac region, whilein the actinide the former barrier height becomes higher than thelatter.24, 25
On the mass-yield curve for the asymmetric fission path, asshown in Figure 4, the light fragment peak of the asymmetricmass-yield curve smoothly moves towards the heavy side when
the nucleus becomes heavier in the actinide region. To the con-

Zhao
trary, the heavy mass peak does not move away as indicated bya vertical dotted line at the fragment mass A ∼ 140. This phenomenon is turned out to be the same reason14 leading to thesymmetric fragment mass valley to fill in (see gross features ofthe bold-dashed curves from Figures 4(C) to 4(D)) at the low excitation energy. Previously, this phenomenon was attributed tothe decreasing probability for the mass asymmetric fission pathwith increasing A f or Z f in actinide region. But the study offission probabilities for the individual fission paths based on the
precise measurements of the TKE intensities indicated that therelative probability for either of the two paths is nearly constantin the light actinide region. This can be also seen in Figure 4,
as the fissioning nucleus becomes heavy, the width of symmetric mass-yield curve becomes narrow and sharp (solid curves),
while the height of peak (vertical arrow with “Symm.” sign) becomes lower. In the same time, the asymmetric peak is lowered.
But this trend was not observed in the region of heavy actinides,
such as in the fission of the 259 Md. The reason has been recently
revealed10 from the deformation properties of the nuclei undergoing individual fission paths: the symmetric fission path in theregion of the heavy actinide is different from the one in nucleiof the light actinides.
5. Conclusion
As the excitation energy varied in such a large range from 10
to 120 MeV, large variations were observed in the characteristics of fission fragments. But the final degree of deformation of
the fissioning nucleus was invariant, giving the β ∼ 1.65 for nuclei in the symmetric fission path and β ∼ 1.53 for those in theasymmetric fission path.
The formulas for the TKE release in the symmetric andasymmetric fission paths, derived from the identical shapeelongation of fissioning nuclei, were presented.
They
are TKE = 0.1173(Z 2f /A1/3f ) + 7 MeV for the former and
TKE = 0.1217(Z 2f /A1/3f ) + 4 MeV for the latter. A comparisonof the presently derived TKE function with the available experimental data, and the results of the dynamical model calculations taking into account the effects of one-body, two-body, andsurface-plus-window dissipations on the dynamical evolution of
the fissioning nucleus beyond the fission saddle point indicatedthat the presently derived TKE formula reproduced well the experimental observations and were in good agreement with the
dynamical calculations for the fission of nuclei in the region of
Z 2f /A1/3
f < 1800. On the mass-yield distribution, it was foundthat the varying trend of the gross characteristics in the massyield curve was qualitatively correlated to the changing trend of
the relative heights of the fission barriers for the symmetric andasymmetric fission paths.
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