Document text
Origin of the narrow, single peak in the fission-fragment mass distribution for 258 Fm
Takatoshi Ichikawa, 1 Akira Iwamoto, 2 and Peter Moller 3
1 RIKEN Nishina Center, RIKEN, Wako, Saitama 351-0198, Japan
2 Japan Atomic Energy Agency, Tokai-mura, Naka-gun, lbaraki 319-1195, Japan
^Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
(Dated: October 28, 2008)
We discuss the origin of the narrowness of the single peak at mass-symmetric division in the fragment mass-
yield curve for spontaneous fission of 258 Fm. For this purpose, we employ the macroscopic-microscopic model,
and calculate a potential-energy curve at the mass-symmetric compact scission configuration, as a function of
the fragment mass number, which is obtained from the single-particle wave-function densities. In the calcu-
lations, we minimize total energies by varying the deformations of the two fragments, with constraints on the
mass quadrupole moment and keeping the neck radius zero, as a function of mass asymmetry. Using the ob-
tained potential, we solve the one-dimensional Schrodinger equation with a microscopic coordinate-dependent
inertial mass to calculate the fragment mass-yield curve. The calculated mass yield, expressed in terms of the
microscopic mass density, is consistent with the extremely narrow experimental mass distribution.
PACS numbers: 24.75.+i, 27.90. +b
In spontaneous fission the fragment mass-yield distribu-
tions change abruptly from a double-peaked, broad, mass-
asymmetric distribution for 256 Fm to a single-peaked, very
narrow, symmetric distribution for 258 Fm JH, 0, [j] . In addi-
tion, in 258 Fm the kinetic-energy distribution can be expressed
as a sum of a low-energy and a high-energy component, whose
mean energies differ by about 35 MeV. The mechanism be-
hind this phenomenon, called bimodal fission, is the strong
nuclear shell effects that appear when symmetric division into
two fragments which both are near the doubly magic nucleus
132 Sn becomes possible. The experimental observations of the
sudden emergence of a mass-symmetric division near 258 Fm
has always been assumed to be due to an emergence in fission
potential-energy surfaces of a fissionpath, strongly stabilized
by fragment shell effects lH |j, S |g, 01 • However, no such in-
tuitive picture of the mechanism behind the narrowness of the
symmetric mass distribution peak has been advanced, nor has
a convincing quantitative calculation explaining the extremely
narrow FWHM been presented.
So far, theoretical investigations have mainly focused on
obtaining the transition point between competing fission
modes near 258 Fm by calculating the potential-energy sur-
face versus various chosen sets of deformation coordinates. In
fact, theoretical models, such as the macroscopic-microscopic
model H, H [13, [H d [Bi me constrained Hartree-
Fock+BCS (HFBCS) model U£ 0], and the constrained
Hartree-Fock-Bogoliubov (HFB) model lfl6l [l7ll . have to a
varying degree of success described such a shell-stabilized
path, and the transition point between asymmetric and sym-
metric fission modes near 258 Fm, that is, the emergence here
of path leading to mass-symmetric divisions with compact
scission configurations, referred to as the compact symmetric
path. Through these investigations the energy-minimum path
leading to high-kinetic-energy, symmetric fission has been
well established, but the structure of the potential valley along
this path and plausible mechanisms behind the extremely nar-
row FWHM have been less extensively studied.
In this paper, we calculate quantitatively the mass distribu-
tion of 258 Fm in the compact symmetric valley. The calcula-
tions are based on studies of the dynamics of the zero-neck-
radius scission configuration in the mass-asymmetry shape
degree of freedom 111 811 . We also impose spherical fragment
shapes, which leaves us with mass asymmetry as the only col-
lective coordinate. This approach implies that the mass distri-
bution originates from zero-point vibrations or thermal fluctu-
ations in the mass-asymmetry degree of freedom at scission.
In the specific case of 258 Fm considered here, it is a reason-
able assumption and approximation, since in spontaneous fis-
sion the exit point after barrier penetration is approximately
the zero-neck scission configuration.
We use the macroscopic-microscopic model lfl9l l20ll
constrained to shapes with zero neck radius. Using a
macroscopic-microscopic model, slightly different from our
implementation, Pashkevich failed to find any significant dif-
ference between the curvatures of the compact and elon-
gated symmetric valleys for 264 Fm 19Q ■ Furthermore, in his
cranking-model analysis of the mass distribution widths asso-
ciated with the zero-point oscillations in his asymmetry de-
gree of freedom he obtained similar mass yield distribution
widths in both valleys, approximately consistent with the nar-
row distribution observed experimentally for 258 Fm. His dy-
namical study has significant similarities with our study here.
However, Pashkevich characterized fragment mass numbers
by the asymmetry of the homogeneous macroscopic volume
defined by the parametrization of the nuclear surface. Here we
use microscopic densities to characterize the mass distribu-
tion, which gives very different mass-distribution curves close
to magic numbers. As we discuss in detail below, we propose
it is the shell gaps that restrict fluctuations in the microscopic
mass-asymmetry degree of freedom.
We use the three-quadratic-surface parametrization I2lll22tl
to describe macroscopic nuclear shapes in a five-dimensional
deformation space. The shape degrees of freedom are a
quadrupole-moment parameter qi, a neck parameter rj, left-
and right-fragment deformation parameters, £\ and £2, re-
spectively, and a mass-asymmetry parameter a g . The pa-
rameter q2 is the dimensionless quadrupole moment in units
of T>ZR\l\% (e 2 b), where Z is the proton number and Rq
2
-20
-30
Elongated symmetric path
Separating ridge
Asymmetric path
Separating ridge
Compact symmetric path
J i i i I i i i I i
6
8 10
92
FIG. 1 : (Color online) Potential energy as a function of the dimen-
sionless quadrupole moment parameter qi- The open squares and
triangles are the minima and optimal saddles obtained by the im-
mersion method in the five-dimensional potential-energy surface, re-
spectively. The solid and dotted lines are the valleys leading to com-
pact and elongated symmetric fission. The dashed line is the valley
leading to the asymmetric fission. The solid and dotted lines with
triangles represent the separation ridge between the compact sym-
metric and the asymmetric valleys and the separating ridge between
the elongated symmetric and the asymmetric valleys, respectively.
is the nuclear radius. The parameter 77 varies from to
1. Scission, with zero neck radius, corresponds to 77 = 0.
The parameter e is the Nilsson perturbed-spheroid parame-
ter. Near scission we have to a very good approximation
a g = (Mi —M2)/(M\ +M2), where M\ and M2 are the vol-
umes of the left and right nascent fragments, respectively. The
microscopic single-particle potential is calculated by folding
a Yukawa function over the macroscopic shape or "sharp-
surface generating volume" lfl9ll .
To study the properties of the compact, mass-symmetric
valley, we calculate the five-dimensional potential-energy sur-
face for 258 Fm and analyze it by use of the immersion method.
Details of the calculation are given in Ref. lf23ll . The parame-
ters correspond to FRLDM(2002) 1I24I1 . For simplicity, we cal-
culate the pairing effect based on the BCS model [25]. Since
we mainly consider scission or near-scission shapes, we can
ignore the shape dependence of the Wigner term. We calcu-
late the potential energies at 41 x20x 15x 15x35 grid points
for Q2, T], Si, £2, and a g , respectively.
Figure 1 shows some main structures identified by immer-
sion techniques in the calculated potential-energy surface for
258p m as a f unC ];i on f g 2 The open squares and triangles are
minima and optimal saddles between minima, respectively.
The energy of the saddle point on the path to compact sym-
metric fission is 0.34 MeV, while that of the asymmetric fis-
sion is 0.58 MeV. The scission point for the compact symmet-
ric fission is at qi = 6.5, £12 = 0, and a g = 0. In the figure,
we show the compact symmetric and the asymmetric valleys,
denoted by a solid and a dashed line, respectively. Those are
separated by a ridge, denoted by a solid line with triangles.
We also find a path leading to elongated mass-symmetric di-
>
CD
*
CD
LU
a>
o
-1— '
i—
CO
CL
I
_CD
CT
CO
o
-t— •
o
i
Q_
80
60
40
258
Macroscopic
Microscopic
Microscopic (with paring)
0.2 0.3 0.4 0.5 0.6
a„
FIG. 2: (Color online) (Upper panel) Proton number of left frag-
ment as a function of the mass-asymmetric parameter a g . The solid
line is the total proton density obtained as a sum over occupied wave
functions in the left fragment. The dashed line is the total proton
density with pairing taken into account. The dotted line is the macro-
scopic left-fragment density. (Lower panel) Nilsson diagram for pro-
ton single-particle levels at scission configurations in the compact
symmetric valley as a function of the mass-asymmetric parameter
Olg. The solid and dashed lines are the energy levels whose wave
function localizes in the right- and left fragments, respectively. The
bold-gray (red) line is the Fermi level.
visions, denoted by a dotted line, but its separating ridge to
the asymmetric valley, denoted by the dotted line with trian-
gles, vanishes at around qi = 10.0, indicating that the scission
point of this path strongly depends on the dynamics after go-
ing through this valley.
Before we discuss the dependence of the potential energy
on mass asymmetry at the compact scission configuration we
need to discuss the relation between fragment mass numbers
and single-particle energy levels. For this purpose, we calcu-
late the proton density for each single-particle state and the
total proton number of the left fragment as a function of a g .
For a g > 0, the volume of the left fragment is greater than
the right fragment. We take E\ and £2 to be in the calcu-
lation. If we maintain our restriction to axially symmetric
shapes, the proton density for the 2-th single-particle state is
given by |y,-(p,z)| 2 , where is the single-particle wave func-
tion in the cylindrical coordinate system. The single-proton
occupation probability in left of two nascent fragments is thus
obtained by
ni = 2n / p\\i/i(p,z)\dzdp,
JO " Zmin
(1)
where z nec k is at a macroscopic neck radius of fm. The
3
FIG. 3: (Color online) Enlargement of part of Fig. 2 with density
plots of a few specific wave functions discussed in the text. In the
insets, the corresponding macroscopic nuclear shape is given by the
solid line.
values p max and z m ; n (negative) are set sufficiently large that
\y/i{p,z)\ 2 becomes negligible outside the integration inter-
vals. The total proton number Zl is the sum of the single-
proton occupation probability from the lowest level to the
Fermi level. We also calculate the total proton number tak-
ing into account the pairing effect, z| BCS \ given by z| BCS ) =
£ vfrii, where vj is the occupation probability calculated using
the BCS pairing model.
We use two density concepts when we discuss the mass
asymmetry in our study. One definition is based on calcu-
lating the mass asymmetry from the single-particle densities,
the other from the asymmetry of the homogeneous volume
defined by the parametrization of the nuclear surface, denoted
"microscopic" and "macroscopic", respectively.
Figure 2 shows important features of our results. The solid
and dashed lines in the upper panel of Fig. 2 show the total
number of protons without and with the pairing effects, Zl
and Z L , respectively. The lower panel of Fig. 2 is a Nils-
son diagram versus a 2 . The thick gray (red) line is the Fermi
level. The energy levels are plotted dashed when more than
half the density is in the left part of the potential, solid other-
wise [see the insets from (a) to (f) in Fig. 3]. The upper panel
shows that in the absence of pairing the total proton number
remains constant at Zl = 50 in the interval < a g < 0.18.
Just below a 2 = 0.20, Zl suddenly jumps to about 60. One
can also see such discontinuities at large a g . Those discon-
tinuities occur where single-particle shell gaps change, that
is they coincide with the level crossing points in the Nilsson
diagram. It is here the downward-sloping states whose wave
-10
I
\
/
/
i .,1 i |i
- * 1
\ \
\ \
. .. 1 .... i i ; i
J I
1
1 /
I /
\
\
\
\
\
\
\
/
/
/
/
/
\ \
\
\
\
\
\
/ /
/ /
/ /
/ /
\ / '
\\
u
/ -
/ -
(c) :
" 258 Fm
i , , , ,
(a)~
, , , , I
(b)
~ Macroscopic
Microscopic
I
45 50 55 75 80 85 120 130 140
Proton Number Z Neutron Number N Mass Number A
FIG. 4: Potential-energy curves at the scission configuration as a
function of the (a) proton, (b) neutron, and (c) total mass densities
of two nascent fragments, respectively. The solid and dashed lines
denote the potential-energy curve calculated using the microscopic
and macroscopic densities, respectively.
functions are localized in the left fragment come below the
Fermi level and can be populated with particles transferring
from the right fragment. These sudden features are smoothed
out when we include the pairing correlation.
In order to see more clearly how Zl increases we show in
Fig. 3 an enlargement of the Fermi-surface region of Fig. 2,
with some wave-function densities inserted. In the figure, the
downward-sloping state denoted by the line (i), coming from
above the Fermi level, dives below the Fermi level. In the case
we have here with paired particles, protons can transfer from
the right fragment into this previously unoccupied state in the
left fragment, which leads to an increase of Zl. In contrast, in
the case the particle number is odd then the odd particle can
not transfer to a level of different £2, in a sudden shape change.
Rather, the odd particle remains in its level until another level
with the same £2 quantum number is encountered, giving rise
to a "specialization energy" and an increase in barrier height.
This is the mechanism behind the long spontaneous fission
half-lives of odd nuclei I26i I27I1 . When a downward sloping
level encounters an upward-sloping level of the same quantum
number £2 when it crosses the Fermi surface, then a repulsion
between the single-particle states occurs. This is the case for
the states denoted by (ii) and (iii), which have the same quan-
tum number, £1 = 5/2, but their wave functions localize on
the right- and left fragments, respectively [see the insets (a)
and (b) in Fig. 3]. The transfer mechanism in the even, paired
system is similar to case (i), but in the single-particle picture
it appears different because when these states come close to
each other the states mix ("level repulsion"), in this case near
a g = 0.185 [see the insets (c) and (d)]. After the mixing re-
gion, the localizations of the wave functions are interchanged
[see the insets (e) and (f)], and the lower level remains below
the Fermi level, but with a new set of quantum numbers. This
microscopic mechanism is responsible for the increase in Zl
with increasing a g .
As shown by the dashed line in the upper panel of Fig. 2, the
pairing interaction smooths the change in the fragment pro-
ton number versus asymmetry, because it incorporates wave-
function admixtures across the single-particle Fermi surface
into the sum of the single-proton densities. As a consequence,
if the density of states just above a shell gap occurring at
4
CA
CO
CD
1 1 r
Inertial Mass
-0.15
0.15
"0
FIG. 5: Inertial mass in the mass-asymmetric direction at the
compact-symmetric scission configuration calculated using the
Inglis-Belyaev formula. The solid lines with squares and circles are
the first term of the RHS in Eq. 3 for neutrons and protons, respec-
tively. The dashed and dotted lines are the second term of the RHS
in Eq. 3 for neutrons and protons, respectively. The solid line is the
total inertial mass for neutrons and protons calculated as the sum of
the first and second terms in Eq. 3.
the Fermi surface is high, then Z L more closely tracks
the macroscopic proton number. At large a g this condition
is particularly well fulfilled and the dashed line is close to
the macroscopic density, denoted by the dotted line in the fig-
ure. However, we found that from a g = to 0. 10, z[ BCS) still
remains constant at 50 due to the very substantial shell gap,
indicating that proton number Z = 50 is extremely stable. We
will show below that this stability may to a large part be the
mechanism that leads the very narrow fragment mass distribu-
tion.
We now calculate the potential-energy curve at the scission
configuration in the compact symmetric valley as a function
of the fragment mass number. In this first study we take £\
and £2 to be zero in calculating the potential energy and the
inertial mass. We have checked this approximation by min-
imizing the total energy with respect to £1 and £2 with con-
straints on (72 = 6.5 and a neck radius of fm, as a g increases.
From oc g = to 0.1, we found that the minimum energy oc-
curs at £1 and £2 equal zero due to the large spherical shell gap,
but abruptly jumps to other valleys near a g =0.1. Inaccura-
cies due to this approximation can therefore be expected to be
fairly insignificant. We calculated the fragment mass number
using the single-particle wave functions with the BCS paring
effect, before this jump takes place.
Figures 4 (a), (b), and (c) show calculated potential energies
as functions of the fragment microscopic proton, neutron, and
total mass densities, respectively. These results are given by
the solid lines. As a comparison, results versus the macro-
scopic density are also given, displayed as dashed lines. The
potential-energy curves versus the microscopic density rise
much more steeply than the curves plotted versus the macro-
scopic density. In particular, the result versus the proton den-
sity depends drastically on the choice of density variable. We
0.15
33
CD
> 0.10
(/>
CO
CO
0.05 -
0.00
1 — 1 — 1 — 1 — 1 — 1 — 1 — 1 — 1 — 1 — 1 — 1 — r
258 Macroscopic
Fm Microscopic
100 120 140 160
Fragment Mass Number A
FIG. 6: Calculated mass-yield curves for compact fission of 258 Fm.
The solid and dashed lines are based on the microscopic and macro-
scopic densities, respectively.
therefore expect that the physical origin of the narrow mass
distribution is the large shell gap. Below we investigate this
hypothesis through a quantitative study.
In order to calculate the mass distribution in the com-
pact symmetric mode, we calculate the zero-point vibration
corresponding to the calculated potential-energy curves dis-
played in Fig. 4, by solving the one-dimensional Schrodinger
equation in terms of a g with a coordinate-dependent inertial
mass B28I1 . For the compact symmetric valley, we expect the
zero-point vibration to be dominating, because the observed
excitation energy of the fission fragments of the nearby 260 Md
in compact, symmetric fission was extremely low II29I1 . The
Schrodinger equation thus reads
1
2VB da g VB da.
+ V(a g )
\ir(a g )=Ey(a s ), (2)
where \jf is the wave function and B is the inertial mass in the
mass-asymmetric direction B aa .
For the inertial mass, we employ the Inglis-Belyaev for-
mula d^HH, given by
Baa = 2tt
< v\dH/da\[l > \ 2 {u v v n +u
where H is the single-particle Hamiltonian, Vm and are the
BCS occupancy and vacancy amplitudes, and is the energy
of the quasi-particle state \ jJ. >. The term P a a gives the con-
tribution from couplings to the pairing vibrations. We use the
finite-difference method to calculate < /i|<9H(o:)/<9a|v >.
Figure 5 shows the calculated inertial mass in units of MqRq,
where Mo = 931 .50 MeV is the atomic mass unit. The solid
lines with squares and circles show the evaluated first term in
the right hand side of Eq. 3 for the neutrons and the protons,
respectively. The dashed and dotted lines show the evaluated
Paa for neutrons and protons, respectively. The solid line is
the sum of those four terms.
We calculate y/(o: g ) i n Eq. 2 using the finite-difference
method. The calculated zero-point energy is —6.05 MeV. The
5
100 120 140 160100 120 140 160
Fragment Mass Number A Fragment Mass Number A
FIG. 7: Mass yields plotted in histogram form. The black-shaded
histograms in the figures (a) and (b) are calculated mass yields us-
ing the microscopic and the macroscopic densities, respectively. The
solid-line histograms are experimental data taken from Ref. 13211 .
calculated ^(^g)' which is the macroscopic density ampli-
tude, is converted into a mass yield function Y(A) in terms
of fragment microscopic densities through, Y(A) — C\ \j/(A)\ 2 ,
where C is a renormalization factor. We chose C so that the to-
tal area of Y is equal to 0.5, because the component originating
from the compact symmetric valley is about 50% of the total
yield. This has been estimated from the two-mode analysis
of the experimental total kinetic-energy distribution The
structure of our calculated potential-energy surface is consis-
tent with this assumption, because the calculated height of
the saddle point leading to the compact symmetric valley is
comparable to that of the asymmetric valley. See also Refs.
Figure 6 shows the macroscopic and microscopic mass-
yield curves. The same data are shown as histograms in
Fig. 7, so that we can compare directly to the experimental
data, which are given in histogram form IB2I1 . In transform-
ing to the histograms, we calculate the average of integrated
values of the mass-yield curve for the 4 u interval used in
the presentation of the experimental data IB2I1 . The solid and
the dashed lines in Fig. 6 correspond to the microscopic and
the macroscopic densities, respectively. The calculated peak
value is consistent with the experimental one, although a one-
dimensional model could have a tendency to overestimate the
peak value of the mass-yield curve |28]. On the other hand, it
is clearly seen that the result of the macroscopic density is not
in agreement with the extremely narrow peak of the experi-
mental mass-yield curve. That is, the single-particle configu-
ration, and specifically the large shell gap, is the source of the
extremely narrow FWHM of the fragment mass-yield curve.
We thus expect that the FWHM of the compact symmetric
component is 3.6 u.
In order to obtain the whole mass distribution, it would be
necessary to clarify valley structures for all fission paths and
superpose their contributions. However, we could not employ
our scission-point model to the other valleys, because it was
not possible to obtain unique scission points for the asym-
metric and elongated symmetric paths. For those paths the
separating ridge vanishes before the scission configurations,
indicating that these components must be modeled in a more
complex dynamical approach.
One may ask how the features we studied here manifest
themselves in self-consistent mean-field calculations which
have also studied 258 Fm. In those models the potential and mi-
croscopic densities are "self-consistent". One could anticipate
that perhaps no self-consistent solutions exist for octupole
constraints corresponding to the range to 0. 18 of a g in Fig. 2.
And when a solution exists there would be a large increase
in energy. If these expectations do occur in self-consistent
models this behavior would tend to very much restrict the
fluctuations in the mass-yield curve, just like in our studies
here. However, in the papers iTfl Iym we find no results that
shed light on precisely these issues. But the calculated bar-
rier versus a quadrupole constraint in lfl4ll is very similar to
the barrier obtained here. Recently [17] looked at correlations
between various fission fragment properties of Fm isotopes.
These calculations do not clearly identify any fragment shell
effect on mass yield widths. This is probably because they
were not specifically designed to study such a possibility. In
Ref. fnll . the Hartree-Fock-Bogoliubov method is used to cal-
culate potential-energy surfaces of Fm isotopes along a scis-
sion line. The results for the 258 Fm potential energy shown in
Fig. 7 have quite different shape compared to our Fig. 4(c). In
particular the potential shows a very shallow minimum cen-
tered at symmetric division. The comparisons of calculated
total kinetic energies to data is not very convincing. The sub-
stantially different results we obtain are likely due to the very
different designs of the two studies, not to the differences be-
tween self-consistent and non-selfconsistent models. In our
case we study mass oscillations near the exit point of compact
scission. In the HFB study energy partitioning along the entire
scission line is the mechanism governing the fragment proper-
ties. Since that mechanism shows substantial differences with
respect to measured data, we feel it is more appropriate to de-
scribe origin of the narrow mass distribution in terms of the
large shell gap at the barrier exit point, as we do here.
In summary, we have presented potential-energy curves of
258 Fm at scission as functions of both macroscopic and mi-
croscopic fragment mass densities. We calculated the zero-
point vibration corresponding to this potential-energy curve
by solving a one-dimensional Schrodinger equation with a
coordinate-dependent inertial mass based on the cranking
model. An important point in the calculation is that the frag-
ment masses are defined by the single-particle wave func-
tions, rather than the macroscopic potential volumes. We
have shown that the mass numbers of two nascent fragments
strongly depend on the single-particle configurations. In par-
ticular, the proton number of the fission fragments originating
from the compact symmetric valley for 258 Fm is strongly con-
strained to Z = 50 due to the large shell gap. The calculated
mass-yield curve is consistent with the extremely narrow ex-
perimental mass yield curve. We obtain that the FWHM of
the fission fragments originating from the compact symmetric
valley is 3.6 u. For 258 Fm, it would be interesting to measure
the ratio of protons to neutrons on the mass yield curve, be-
cause the neutron distribution may be wider than the proton
distribution, since the mean fragment neutron number is not
magic. This would be a very strong test of the mechanism
6
behind the narrow mass distribution.
Acknowledgments
TI is grateful for the Special Postdoctoral Researcher Pro-
gram in RIKEN. The numerical calculations have been per-
formed at the RSCC system, RIKEN. PM would like to ac-
knowledge that this work was carried out under the auspices
of the National Nuclear Security Administration of the U.S.
Department of Energy at Los Alamos National Laboratory
under Contract No. DE-AC52-06NA25396 and was also sup-
ported by a travel grant to JUSTIPEN (Japan-U.S. Theory In-
stitute for Physics with Exotic Nuclei) under grant number
DE-FG02-06ER41407 (U. Tennessee).
[1] D. C. Hoffman, J. B. Wilhelmy, J. Weber, W. R. Daniels, E. K.
Hulet, R. W. Lougheed, J. H. Landrum, J. F. Wild, and R. J.
Dupzyk, Phys. Rev. C 21, 972 (1980).
[2] E. K. Hulet, J. F. Wild, R. J. Dougan, R. W. Lougheed, J. H.
Landrum, A. D. Dougan, M. Schadel, R. L. Hahn, P. A. Bais-
den, C. M. Henderson, et al., Phys. Rev. Lett. 56, 313 (1986).
[3] E. K. Hulet, J. F. Wild, R. J. Dougan, R. W. Lougheed, J. H.
Landrum, A. D. Dougan, P. A. Baisden, C. M. Henderson, R. J.
Dupzyk, R. L. Hahn, et al., Phys. Rev. C 40, 770 (1989).
[4] Y. L. Zhao, I. Nishinaka, Y. Nagame, M. Tanikawa, K. Tsukada,
S. Ichikawa, K. Sueki, Y. Oura, H. Ikezoe, S. Mitsuoka, et al.,
Phys. Rev. Lett. 82, 3408 (1999).
[5] Y. L. Zhao, Y. Nagame, I. Nishinaka, K. Sueki, and H. Naka-
hara, Phys. Rev. C 62, 014612 (2000).
[6] H. C. Britt, D. C. Hoffman, J. van der Plicht, J. B. Wilhelmy,
E. Cheifetz, R. J. Dupzyk, and R. W. Lougheed, Phys. Rev. C
30,559 (1984).
[7] D. C. Hoffman and M. R. Lane, Radiochim. Acta 70/71, 135
(1995).
[8] P. Moller, J. R. Nix, and W. J. Swiatecki, Nuclear Physics A
469, 1 (1987).
[9] V. V. Pashkevich, Nucl. Phys. A 477, 1 (1988).
[10] P. Moller, J. R. Nix, and W. J. Swiatecki, Nuclear Physics A
492, 349 (1989).
[11] S. Cwiok, P. Rozmej, A. Sobiczewski, and Z. Patyk, Nucl. Phys.
A 491, 281 (1989).
[12] P. Moller and J. R. Nix, J. Phys. G: Nucl. Part. Phys. 20, 1681
(1994).
[13] P. Moller, D. G. Madland, A. J. Sierk, and A. Iwamoto, Nature
(London) 409, 785 (2001).
[14] L. Bonneau, Phys. Rev. C 74, 014301 (2006).
[15] A. Staszczak, J. Dobaczewski, and W. Nazarewicz, Acta Phys.
Pol. B 38, 1589 (2007).
[16] M. Warda, J. L. Egido, L. M. Robledo, and K. Pomorski, Phys.
Rev. C 66, 014310 (2002).
[17] N. Dubray, H. Goutte, and J.-P. Delaroche, Phys. Rev. C 77,
014310 (2008).
[18] R. Vandenbosch and J. R. Huizenga, Nuclear fission (Academic
Press, INC., London, 1973).
[19] M. Bolsterli, E. O. Fiset, J. R. Nix, and J. L. Norton, Phys. Rev.
C 5, 1050 (1972).
[20] P. Moller, J. R. Nix, W. D. Myers, and W. I. Swiatecki, At. Data
Nucl. Data Tables 59, 185 (1995).
[21] J. R. Nix, University of California Radiation Laboratory Report
UCRL-17958 (1968).
[22] I. R. Nix, Nucl. Phys. A130, 241 (1969).
[23] P. Moller, A. J. Sierk, T. Ichikawa, A. Iwamoto, R. Bengtsson,
H. Uhrenholt, and S. Aberg, to be published.
[24] P. Moller, A. J. Sierk, and A. Iwamoto, Phys. Rev. Lett. 92,
072501 (2004).
[25] P. Moller and I. R. Nix, Nucl. Phys. A 536, 20 (1992).
[26] I. O. Newton, Prog. Nucl. Phys. 4, 234 (1955).
[27] I. Randrup, C. F. Tsang, P. Moller, S. Nilsson, and S. E. Lars-
son, Nucl. Phys. A217, 221 (1973).
[28] P. Lichtner, D. Drechesel, J. Maruhn, and W. Greiner, Phys.
Lett. 45B, 175 (1973).
[29] I. F. Wild, I. van Aarle, W. Westmeier, R. W. Lougheed, E. K.
Hulet, K. J. Moody, R. I. Dougan, E.-A. Koop, R. E. Glaser,
R. Brandt, et al., Phys. Rev. C 41, 640 (1990).
[30] M. Brack, I. Damgaard, A. S. lensen, H. C. Pauli, V. M. Struti-
nsky, and C. Y. Wong, Rev. Mod. Phys. 44, 320 (1972).
[31] I. Randrup, S. E. Larsson, P. Moller, S. G. Nilsson, K. Po-
morski, and A. Sobiczewski, Phys. Rev. C 13, 229 (1976).
[32] E. K. Hulet et al. :Data file EXFOR-13191.002 dated 1998-08-
01, compare Phys. Rev. C 40, 770 (1989). EXFOR data received
from the IAEA Nuclear Data Section, Vienna.