Document text
(N
Selfconsistent calculations of fission barriers in the Fm
region.
M. Warda^'^ J. L. Egido\ L. M. Robledo\ and K. Pomorski^'^
^Departamento de Fisica Tedrica C-XI, Universidad Autdnoma de Madrid,
Madrid, Spain
O ; '^Katedra Fizyki Teoretycznej, Uniwersytet M. C. Sklodowskiej,
CN ! Lublin, Poland
c5 ■ ^IReS - IN2P3 - CNRS and Universite Louis Pasteur,
Strasbourg, France
(N
T— I
^ ' Abstract
[p^ I The fission barriers of the nuclei ^^'^Fm, ^^^Fm, ^^^Fm, ^^^No and ^^'^Rf are investi-
' gated in a fully microscopic way up to the scission point. The analysis is based on the
. constrained Hartree-Fock-Bogoliubov theory and Gogny's DIS force. The quadrupole,
i octupole and hexadecapole moments as well as the number of nucleons in the neck re-
' gion are used as constraints. Two fission paths, corresponding to the bimodal fission, are
found. The decrease with isotope mass of the half-life times of heavy Fm isotopes is also
explained.
>
: PACS numbers: 21.60.Jz,21.10.Dr,21.10.-k,21.10.Pc
1 Introduction
Due to the loss of stability, with respect to spontaneous fission, the number of elements is
limited to only few more than a hundred and ten. Both the experimental and theoretical studies
of spontaneous-fission properties are crucial for understanding the stability properties of the
heaviest elements. The abrupt transition that occurs from ^^^Fm to ^^^Fm in fission-fragment
mass and kinetic-energy distributions, and in the spontaneous fission half-lives of heavy nuclei
was found experimentally (see e.g. the review articles [jl], 0]). For ^^^Fm and heavier isotopes,
the spontaneous-fission half-life decreases relative to ^^^Fm by several orders of magnitude.
The mass distribution of fission fragments of ^^^Fm becomes very narrow with a single peak
at symmetrical fission and the kinetic-energy distribution has two peaks: one at high energy
(230 MeV) and the second less prominent at lower energy (205 MeV). On the other hand, the
256pj^ isotope exhibits a rather strong mass asymmetric distribution Al/Ah = 112/141 and
only the low energy peak is observed in the kinetic energy distribution. This is a rather puzzling
1
situation as it is not expected, from a macroscopic point of view, a substantial change in the
properties of both isotopes. Therefore, the different fission properties of both isotopes have to
be attributed to subtle shell effects making its theoretical explanation even more challenging.
The qualitative explanation of all these phenomena by the existence of an additional fission
valley in the multidimensional potential energy surface was proposed by Hulet et al. [|, |].
They assumed a bimodal character of the kinetic-energy and mass distributions to show that
for ^^^Fm there should exist two different fission paths leading to two distinctly different scission
configurations. The first one is the conventional scission configuration of two fairly elongated
shapes corresponding to the low kinetic-energy peak and the broad mass distribution. The
second one leading to compact scission, i.e. configuration of two touching spheres correspond-
ing to the high energy peak in the kinetic energy distribution and the narrow peak in mass
distribution at symmetric fission.
After the discovery of bimodal fission in ^^^Fm P, 3, a number of theoretical papers has
focused on this problem [^, ||, |^, ||, ^ [1^, |lT|. All these papers are based on the mean-field
single-particle potential and the Strutinsky shell correction method. Most of them deal with
the form of the potential energy surface only. These static calculations usually give two fission
valleys: one leading to the elongated form of fission fragments (EF) and the second one which
corresponds to two nearly spherical fragments, which is usually referred as the compact fission
(CF) valley. It has been pointed out by Brosa and others that this new CF valley is associated
with the doubly magic (Z=50, N=82) shell closure in the fission fragments.
The macroscopic-microscopic calculations of the potential energy surface (PES) for ^^^Fm
reported in Ref. p are based on the Woods-Saxon single particle Hamiltonian. The collective
potential energy surface V{f3, (34,) was minimised there with respect to the deformation param-
eters jS^, jS^ and /Sq. Two fission valleys were found, bifurcating right after the exit from the
fission barrier. It was also shown in [Q that for the largest values of f3 {(3 = 1.6) the EF valley
correspond again to symmetric fission {(3^ = (3^ = 0). This theoretical result is in line with
the observation of Hulet et al. [0, ^ where they found symmetric fission only and equal fission
half-life times for both fission modes, i.e. for the fragments with the large total kinetic energy
(CF valley) and for those with the small total kinetic energy (EF valley).
Contrary to the estimates of Ref. 0, the results obtained by Moeller et al. @, |^, |1^ indicate
the existence of the second barrier on the EF path. The authors have found even a third switch-
back path going from the SF valley via the second saddle to the EF valley. These calculations
are based on the finite-range liquid drop model and the folded- Yukawa single particle potential.
Additionally, in Refs. |^ a smaller mass parameter is postulated along the CF path in order
to obtain the comparable spontaneous fission half-life time for both fission modes of ^^^Fm. In
Refs. |TD|, O] Moeller et al. tried to find a solution to this problem by making calculations
on a 5-dimensional space using a finite range liquid droplet model. They have also found two
paths leading to fission, but the low energy mode was presented as an asymmetric one, with
mass asymmetry Mh/Ml = 152.2/105.8, which does not agree with the experimental data. It
was shown in |P, Q that in ^^^Fm both modes of fission only lead to a symmetric split of the
nucleus.
There are also some estimates of fission barriers made within the constrained Hartree-Fock-
Bogoliubov (HFB) approximation with Gogny or Skyrme effective interactions both at zero spin
1|, |1|, |15|, 0, [13, |18|, |19|, 13, m and at high spin [||] . Most of these calculations where made for
nuclei with Z < 100, where only the traditional fission path (EF) appears in the experimental
data. In |21] calculations for Z > 100 showing the compact fission path (CF) where carried out
for both non-relativistic (Skyrme interaction with the SkI4 parameters) and relativistic (in the
2
context of the relativistic mean field with the PL-40 parameterization) frameworks.
Recapitulating we can say that nuclei in the Fm region represent a very wide variety of fission
types. The low mass isotopes fission with an asymmetric mass distribution of the fragments.
The spontaneous fission half-life (T^/) increases from about 0.8 ms for ^^^Fm up to 126 y for
252pj^ and then falls down again to 0.36 ms for ^^^Fm in order to grow again by one order of
magnitude for ^^°Fm. The situation changes especially dramatically when one goes from ^^^Fm
to ^^^Fm. The fission half-life decreases by 7 orders of magnitude and reaches 0.36 ms and, as
was described above, a very narrow symmetric mass distribution appears in the fission yield.
Moreover, a similar behavior characterizes heavier nuclei in the neighborhood of ^^^Fm. Up to
now this rapid change of systematics of Tsf has not been well described theoretically |jlT|, ^ .
The aim of the present investigation is to look at the form of the fission barriers obtained
with Gogny forces for the nucleus ^^^Fm and its neighbors. In particular our aim is to answer
the following questions:
- do there exist two fission valleys for nuclei in this region?
- is it possible to reproduce the experimental mass distributions of the fission fragments?
- is it possible to explain the rapid change of the measured fission half-life between ^^^Fm
and 258Fm?
In order to answer the above questions we have performed constrained Hartree-Fock-Bogoliubov
calculations as described in Sect. 2, where we give a brief outline of the theoretical model as
well as the description of the forces and the configuration space. In Sect. 3 we discuss the
results obtained. Sect. 4 contains a summary and some concluding remarks.
2 Theoretical model
The Gogny density- dependent effective nucleon-nucleon force is taken in the following form [^]:
(fi -
i=l
+ i VTls (Vi - V2) X 5{fi - f2)(Vi - V2) ■ {ai + aa) (1)
+ to (1 + xoP<x) 5(fi - fa)
P( — ^ — )
+ Vcoul
which contains a central finite range interaction, a zero-range spin-orbit term and a zero-range
density dependent interaction, respectively. The Coulomb interaction has to be added in the
case of protons. The central interaction is a sum of two Gaussian with widths /ii and ^2- P<^
and Pt denote the spin and isospin exchange operators respectively, and p is the total density.
We use the DIS |T^, ^ parameterization of the Gogny interaction. The DIS parameters
were adjusted [|T^ to give a better surface energy term (crucial for a proper description of the
fission phenomenon) and their numerical values are given by:
Wi = -1720.30 MeV W2 = 103.639 MeV
Bi = 1300.00 MeV B2 = -163.483 MeV
Hi = -1813.53 MeV H2 = 162.812 MeV
3
Ml = 1397.60 MeV M2 = -223.934 MeV (2)
/ii = 0.7 fm /X2 = 1-2 fm
to = 1390.6 MeV fm^(^+^) xo = 1
7= 1/3 Wls= 130 MeV fm^
The choice of the Gogny force with the DIS parameterization is based on the fact that whenever
this interaction has been used to describe low energy nuclear structure phenomena an, at least,
reasonable agreement with experiment has always been obtained. This degree of agreement has
been obtained both for calculations at the mean field level and beyond. |T3|, |T^, |TB|, ^
m m m, m m, isg, psi, pi, br pe
In the microscopic HFB calculations we have used the computer code of [p9l where special
attention was paid to an accurate computation of the matrix elements of the Gogny interaction
for very big bases like the ones used in this paper. The self-consistent equations have been
solved by expanding the quasiparticle creation and annihilation operators on finite bases of
axially symmetric deformed harmonic oscillator (HO) eigenfunctions. The size of the bases
used depend upon two parameters, Nq and q, which are related to the allowed range of the HO
quantum numbers trough the relation
-n^ + (2n^ + Iml) < A^o-
Q
Along the perpendicular direction we take A^o shells, (i.e. 2n± + \m\ = 0, . . . , A^o) and along the
z direction we include up to gA'o shells depending on the value of 2n± + \m\. In the present
study we have used q = 1.5, a value which is suited for the elongated shapes along the z
direction typical of the fission process, and A"o = 13, 15, and 17. The reason to use different
values of A'q is to study the convergence of our results with the basis size. Another parameters
characterizing the HO bases are the oscillator lengths b± and bz- These two quantities have
been determined, for each calculated wave function, as to minimize the HFB energy for the
A'o = 13 basis. The same values of b± and bz are then used in subsequent calculations with
A'o = 15 and 17 (see below for a discussion of the convergence).
To study triaxiality effects in the first fission barrier we have also carried out calculations
where the axial symmetry requirement was released but the left-right symmetry was imposed.
As these calculations are much more time consuming than the axially symmetric ones we had
to restrict them to the A"o = 13 case but, as it will be discussed later, this is not a limitation
in the region of interest.
In order to study the different paths to fission we have used in our calculations the following
constraints: the axial quadrupole (Q2), octupole (Qs) and hexadecapole (Q4) moments as well
as the number of nucleons in the neck region (Qn)- The corresponding operators are given by:
Qx = r^Pxicosi9)) and Q^v = exp ( ^ ) , (3)
\ Oat /
with aAr=l fm.
In the minimization process neither the two body kinetic energy correction nor the Coulomb
and spin orbit pairing energies have been taken into account. Additionally, the Coulomb
exchange energy has been treated in the Slater approximation [^D|, The reasons are the
following: First, the calculation of the Coulomb exchange and pairing energies is extremely time
consuming ||35[ and its inclusion would prevent the large scale calculations presented in this
4
paper. From we know that Coulomb pairing can be very important for collective masses but
has little influence in the energy landscape. On the other hand, the Slater approximation to the
Coulomb exchange energy works fairly well in all the cases (spherical or deformed nuclei) and is
an affordable and reliable approximation. Concerning the spin-orbit pairing its contribution to
the pairing fleld is very small, specially at zero spin, and can be safely neglected. Finally, the
two body kinetic energy correction (2b-KEC) is not included in the variation process because,
for heavy nuclei, it remains almost constant for most of the physical conflgurations. As this
term was included in the fltting of the force, we have to include its contribution at the end of
the calculation in order to obtain reasonable binding energies.
We have also subtracted from the HFB energy the rotational energy (REC) corrections
stemming from the restoration of the rotational symmetry. This correction has a considerable
influence on the energy landscape (and therefore on the height of the flssion barriers) as is
somehow proportional to the degree of symmetry breaking and therefore proportional to the
quadrupole moment. A full calculation of the REC would imply an angular momentum projec-
tion ||36|, ^ which is only feasible for light nuclei. In order to estimate the REC we have followed
the usual recipe of subtracting to the HFB energy the quantity (Aj2)/(2Jy), where (AJ^)
is the fluctuation associated with the angular momentum operators in the HFB wave function
and Jy is the Yoccoz moment of inertia This moment of inertia has been computed using
the "cranking" approximation in which the full linear response matrix appearing in its expres-
sion is replaced by the zero order approximation. The effect of the "cranking approximation"
in the Yoccoz moment of inertia was analyzed with the Gogny interaction for heavy nuclei in
by comparing it with the one extracted from an angular momentum projected calculation
(see also [|3^ for a comparison in light nuclei). The conclusion is that the exact REC is a factor
0.7 smaller than the one computed with the "cranking" approximation to the Yoccoz moment
of inertia for strongly deformed conflgurations (a similar behavior has been observed for the
Thouless-Valatin moment of inertia in |^). We have taken this phenomenological factor into
account in our calculation of the REC.
In the last section we analyze the spontaneous flssion half life of several Fm isotopes. The
analysis was carried out in the standard WKB framework where Tgf is given (in seconds) by
Tsf = 2.86 • 10^2^(1 + exp(25)) . (4)
In this expression S is the action along the constrained path which is given by
S= t dQj2B{Q2){y{Q2)~E,) . (5)
J a
For the collective quadrupole inertia B{Q2) we have used the ATDHFB expression computed
again in the "cranking" approximation and given by ||4l[
with
Batdhfb{Q2) = , (6)
Here is the 20 component of the quadrupole operator Q2 in the quasiparticle representation
4^ and E^^ are the quasiparticle energies obtained in the solution of the HFB equation.
5
In the expression for the action the collective potential V{Q2) is given by the HFB energy
(with the 2b-KEC and REC corrections) minus the zero point energy (ZPE) correction eo{Q2)
associated with the quadrupole motion. This ZPE correction is given by
(8)
where
G{Q2)
M_2{Q2)
(9)
Finally, in the expression for the action and additional parameter Eq is introduced. This
parameter can be taken as the HFB energy of the (metastable) ground state. However, it is
argued that in a quantal treatment of the problem the ground state energy is given by the HFB
energy plus the zero point energy associated to the collective motion. To account for this fact,
the usual recipe is to add an estimation of the zero point energy to the HFB energy in order to
obtain E^. In our calculations we have taken a zero point energy of 0.5 MeV for all the isotopes
considered.
2.1 Convergence of the calculations
In our calculations bases with Nq = 13, 15 or 17 were used in order to check the convergence of
the results. As an example of these tests we display in Fig. 1 the HFB energies corresponding
to the CF path of ^^^Fm as a function of Q2 for different values of N^. A comparison of the
Nq = 13 and the A^o = 15 results show that Nq = 13 is enough in the region around the first
barrier but this is not the case in the region of the superdeformed minimum, around Q2 = 100 b,
where a 4 MeV shift is observed in going from Nq = 13 to Nq = 15.
Using Nq — 17 we obtain rather stable results as compared with the Nq — 15 calculations.
That is, for most of the Q2 range the difference between the A'^o = 17 and A'o = 15 energies is
almost independent for Q2. The difference becomes visible only for Q2 > 200 b, but this region
is irrelevant to our investigation as it corresponds to solutions with well separated fragments.
This behavior is typical for all paths and nuclei presented in this paper and from the above
considerations, one can conclude that A^o — 15 is sufficient for the planned HFB calculations.
We have also checked that this fast convergence with the basis size is a consequence of the op-
timization of the oscillator lengths carried out for each quadrupole deformation. The oscillator
length parameters were chosen as to minimize the HFB energies in the A^o = 13 calculations.
2.2 Two body kinetic energy and rotational energy corrections
The influence of the two-body kinetic energy (2b-KEC) and the rotational energy (REC) cor-
rections on the binding energy and the fission barrier of ^^^Fm is shown in Fig. 2. It is seen
in the figure that the 2b-KEC shifts up the binding energy by around 13 MeV with respect
the HFB estimate, while the REC is negative and its magnitude grows from for the spherical
configuration to about 5 MeV in the region of the second barrier (see inset). The plateau
observed in the REC starting in Q2 = 140b is due to the fact that from there on the solution
correspond to two separated spherical fragments. In this case both the moment of inertia and
the fluctuation of the angular momentum operators are proportional to the square of the dis-
tance between the fragments. As a consequence of the behaviour of the REC as a function of
the quadrupole moment, its inclusion decreases the second barrier by a few MeV and that has
6
an important influence on the systematics of tlie spontaneous fission life-times of heavy Fm
isotopes as it will be seen in the next section. All potential energy surfaces (PES) presented in
this paper contain both the 2b-KEC and REC corrections described above.
3 Results
All nuclei considered here have similar barrier shapes exhibiting two humps. The ground
state minimum is at approximately (^2=15 b, which corresponds to the deformation parameter
/?2 ~ 0.25. The first fission barrier is, in our axially symmetric calculations, about 10 MeV high
but it is decreased by a few MeV when triaxial shapes are included in the analysis what is in
agreement with results of other authors (consult e.g. Refs. p^ , |21[|). The lowering of the fission
barrier due to triaxiality comes together with an increase of the collective mass and therefore
the effect of triaxiality on the fission half-lives is rather small. This is in agreement with the
results of Ref. p4| where it was found that the least action trajectory, or in other words the
dynamical path to fission, leads only through the axially symmetric shapes of fissioning nucleus.
Finally, a superdeformed minimum at an energy similar to the ground state energy appears at
Q2 ~ 50 b. It is separated from the scission point by a small second barrier that, as we will see
in the next section, plays a fundamental role in the fission half-lives.
3.1 Nuclear Properties along the fission paths
In this section the potential energy the octupole and hexadecapole Q4 moments as well
as the neck parameter are investigated along the fission paths for the nuclei 256,258pj^^ ^ssj^Jq
and 260Rf.
3.1.1 256p^_
In Fig. 3, the results of the calculations for the nucleus ^^^Fm are plotted for the CF path (solid
lines) and the EF path (dashed lines). In panel (a) we show the potential energy surfaces as
a function of the quadrupole moment for both paths. Along with the energy curve we have
also plotted the real shape of the nucleus for relevant values of the quadrupole moment. The
reduction of the first barrier by approximately 4 MeV due to the triaxial degrees of freedom
(7 is typically in the range between and 8 degrees) is marked by the dotted line. We observe
that after tunnelling through the first barrier the nucleus goes into the superdeformed region.
In fact there are two superdeformed minima, one at Q2 =50 and another at 70 b separated by
a tiny barrier. The deeper minimum at Q2 = 70 b is situated 2 MeV below the ground state
energy. However, it is only separated from the scission point at Q2 = 130 by a barrier which
is only 2 MeV high and therefore it is rather unlikely that this superdeformed (ground state)
minimum can live long enought as to be considered a metastable state. The fission products
corresponding to this path are identical and spherical, in fact, the fragments are two spherical
^^^Sn nuclei. Such type of fission path (solid line in Fig. 3) was called in Ref. |^ the compact
fission (CF) path. It was also shown in p| that the octupole moment along such a path is equal
to zero what is in line with our results, see panel (b) of Fig. 3. After passing the scission point
the potential energy (the Coulomb energy in fact) of this fissioning system decreases rapidly
with growing quadrupole moment. The other path, called elongated fission (EF) path, begins
at (^2=70 b. This path plays a crucial role in the fission process of this nucleus. It corresponds
to the reflection asymmetric shapes with Q3 7^ as one can observe in panel (b) of Fig. 3.
7
Both fission paths differ also significantly in the hexadecapole moments, panel (c), and in the
number of nucleons in the neck region, panel (d). For quadrupole moments larger than 120 eb
one finds that the EF path has a gentler slope than the CF path.
In order to understand the shapes of the EF path for this nucleus and the heavier isotopes
considered we have plotted in panel (a) of Fig. 4 the shape of ^^^Fm at the deformation
Q2 = 200 b. On the left-right asymmetric shape distribution of the fissioning nucleus one can
distinguish two fragments connected by a neck. One of the fragments is close to a sphere and
the other one has a rather large quadrupole deformation. In order to study the mass contents
of both fragments we have plotted in panel (b) the quantity
/Z POO
dz' / dr^p{rxz')
-00 Jo
for both protons and neutrons. The number of particles corresponding to the magic numbers
50 and 82 are marked in the panel (b) with horizontal dotted lines. From this plot we learn
that both fragments have roughly the same mass and they correspond to Sn isotopes close to
the doubly magic ^^^Sn. The fact that an strongly left-right asymmetric mass distribution leads
to two fragments with roughly the same mass is a remarkable result that will be commented
later. Finally, no significant lowering of the density is observed in the neck region (panel (c) of
Fig. 4).
The transition from the CF to the EF path take place first at Q2 = 90 b because, for smaller
quadrupole moments, the paths are separated by a 5 MeV high ridge as can be seen in Fig. 5,
where cross sections of the potential energy surface for various quadrupole moments are plotted
as a function of the neck parameter Qn- When the barrier between both valleys disappears the
nucleus continues fission along the EF path. Such a behavior, referred to by other authors a
"switchback path" |T2[ seems to be energetically most preferable. From Q2=100 b up to 130 b
the minimum corresponding to the CF valley becomes a shoulder as it is seen in Fig. 5. This
means that Fill can not continue fissioning along this mode and will proceed through the
EF path explaining the low kinetic energy distribution of the fission fragments of this nucleus.
The minimum corresponding to the CF valley appears again at Q2=140 b but at such a large
quadrupole moments both fragments are already separated. At (52=140 b the fission valleys
CF and EF are separated by a 4 MeV high barrier.
The fragments which are created in the EF process of ^^^pj^i have different deformations
but nearly equal masses as seen in Fig. 4. This is inconsistent with the experimental mass
distribution which shows a mass asymmetry Ah/Al ~ 141/112 [^. Our static calculations are
based only on the PES of the fissioning nucleus and it seems that dynamical effects could play
a certain role in the fission of ^^^pj^i. it could also happens, but this is rather less probable,
that we are not able to find such a mass asymmetric path in our calculation. Apart from the
two valleys described above a few others paths were found. All of them are localized much
above the CF and EF paths so it is rather improbable that the fissioning nucleus will follow
one of them.
3.1.2 258pj^_
It was found experimentally in Ref. 0, Q that the nucleus ^^spm exhibits bimodal fission.
Both modes have similar abundance and symmetric mass distribution. In panel (a) of Fig. 6
the CF (solid line) and EF (dashed line) fission barriers are shown. The octupole, hexadecapole
moments and the number of nucleons in the neck region Qn corresponding to both paths are
8
presented in panels (b-d) of Fig. 6. Generally speaking the picture is very similar to the one
of ^^^Fm, but there are some important differences which cause changes in the fission yield of
The first distinction between this nucleus and ^^^Fm is the fact that the second hump of
the fission barrier on the CF path (at (52=120 b) has practically disappeared and rises only 0.5
MeV above the superdeformed minimum. Additionally the top of the second barrier is placed
a few MeV below the ground state energy.
The second difference with respect to ^^^Fm is the relation between the two paths leading
to fission. As it is seen in Fig. 7, the minima corresponding to the CF and EF coexist along
the fission path, i.e., the ridge between them does not disappear along the whole way to fission.
Its height always exceeds 1.5 MeV. It means that the nucleus could fission via the CF valley or
change the path and proceed with the elongated fission (EF) fragment path starting from Q2 ~
90 b. At higher quadrupole moments the transfer from the CF to the EF path is also possible
but it is less probable as the ridge between the paths in the region of the second hump rises up
to 3 MeV. Such a configuration of the EF and CF paths seems to ensure (because we do not
account here for dynamical effects) that both modes are fed with similar intensity. This result
is in agreement with conclusions of Refs. 0, ^ , where comparable abundance of both modes
was found.
We can identify the CF path with the high total kinetic energy (TKE) mode in the fission
of ^^'^Fm and the EF path with the low TKE mode. In the CF path the nucleus splits into
two identical, spherical parts which are two ^^^Sn nuclei. At the scission point, the distance
between the centers of masses of these spherical fission fragments is relatively small what gives
a strong Coulomb repulsion and in consequence a high mean value of the TKE of the fragments.
The fissioning nucleus passing through the second EF path has a similar elongated shape as
the one described for ^^^Fm in Fig. 4. The distance between the mass centers of the two born
fragments is much larger than the one for the CF path. It causes a weaker Coulomb repulsion
between fragments and in consequence a smaller mean TKE of the fragments.
We have got also some arguments in favor of the hypothesis that in the fission of ^^^Fm
one deals with a kind of cluster fission 0, both in the CF and in the EF paths. Looking at
the proton and neutron density distribution along both valleys we have found that the nucleus
258pj^ splits into two parts with equal masses. Each fragment has around 50 protons and
79 neutrons. The only difference is that one of the ^^^Sn fragments born in the EF path is
highly deformed with /32 = 0.6 whereas is spherical in the CF path. As the different TKE of
the two paths could be explained in terms of the energy difference between the spherical and
the superdeformed fragment, we have performed additional (52-constrained HFB calculations
for a few Sn isotopes. The results for ^^^^^^^Sn even-even isotopes are presented in Fig. 8.
A shoulder (or even fiat minimum for ^^'^Sn) is seen for all isotopes at (52=10 to 14 b. This
superdeformed second minimum for ^^°Sn corresponds to (^2=12 b (or (32 = 0.6) and it is
located around 23 MeV above the ground state. This superdeformed state can be identified
as the deformed ^^^Sn fragment meaning that the 23 MeV accumulated in the superdeformed
state will be taken away by post-fission neutrons or 7-rays and will not be converted to kinetic
energy of the fragments. Therefore, we expect that the TKE of the fragments for the EF path
to be of the order of 23 MeV smaller than for the CF path, in good agreement with experiment.
9
3.1.3 258^0 and ^eoRf.
Figs. 9 and 10 (Figs. 11 and 12) show the PES and its cross sections for the fissioning ^^®No
(^^°Rf ) nucleus. In these nuclei, the second hump of the potential barrier disappears completely.
After tunneling through the first barrier they fission directly.
In the nuclei ^^sNo (Fig. 9) and ^eo^f (pig. H), as in ^sspm, the CF and EF paths are
also found. The transition between both valleys is possible at (^2=90 b, as it can be seen in
Figs. 10 and 12, where there is no ridge separating them. Here, similarly to the case of ^^^Fm,
the minimum corresponding to the CF path turns into a shoulder at about (^2=100 b, but it
appears again at Q2=120 to 130 b. It makes possible the come-back to the CF path, although
the probability for such a process is relatively small. This effect explains the experimentally
observed low abundance of the high TKE mode: 5% for ^^^No and even less for ^^°Rf.
In configurations close to scission another third path corresponding to two compact frag-
ments (marked by the dotted lines in Figs. 9 and 11) will appear. In contrast to the above
considered Fm isotopes, the two fragments have a small asymmetry in mass with a rate of
Mh /Ml=132.5/125.5 for ^s^No and Mh /Ml=13Q/12A for ^^^RL In both cases the number of
protons is the same in both fragments. This asymmetry is in line with the experimental results
for these nuclei [Q, |].
3.2 Spontaneous fission half-lives
For heavy Fm isotopes the spontaneous fission half-lives (Tsf) decrease rapidly with mass. This
fall off has up to now (see e.g. |2^) not found a satisfactory explanation.
The shape of the potential barrier is one of the most important factors which determines
the fission half-life of nuclei. The fission barriers for ^^'^Fm, ^^^Fm and ^^^Fm (for the CF path
only) are plotted in Fig. 13. All curves are shifted in order to get the ground state minimum
in the same position. One can see in Fig. 13 that the first hump of the barrier is practically
the same for all these nuclei. In fact, the almost 10 MeV high fission barriers are reduced by
a few MeV when including the effect of triaxiality (see in Figs. 3 and 6). However, as already
mentioned, it was shown in Ref. that the dynamical effects prevents the fissioning nucleus
to take axially non symmetric forms, so we have decided to perform the estimates of Tgf for
the both (axial and nonaxial) cases.
The main difference between these three isotopes is in the location of the second hump with
respect to the ground state minimum. In ^^^Fm the second barrier is at the ground state level
and in ^^^Fm a few MeV lower than that. In these nuclei only the first hump will influence the
fission half-life time. The fission barrier of ^^^Fm has a completely different shape. Although,
the shift up of the second hump with respect to ^^^Fm is not larger than the corresponding shift
in ^^^Fm with respect to ^^^Fm, this small shift causes that the whole second barrier is now
above the ground state. This effect influences the theoretical estimates for flssion half-lives in a
dramatical way. The half-life for ^^^Fm is in our estimates 11 orders of magnitude larger than
that for ^^^Fm when the symmetric (CF) path (solid line in Fig. 3) is taken into account (open
symbols in Fig. 14). It becomes much shorter (full symbols in Fig. 14), and almost equal to
the experimental one, when the reduction of the flssion barrier due to the left-right asymmetry
degrees of freedom is included. It has obviously to do with the fact that the size of the flssion
barrier to be tunneled in ^^^Fm reduces slightly if one switches to the EF valley at Q2 ~90 b.
The estimates done assuming the axially symmetric form of flssioning nucleus are marked in
Fig. 14 by the circles, while these for the nonaxial case are denoted by the triangles. It is seen
10
that the inclusion of the nonaxial degrees of freedom decreases the values of T,/ obtained in
our one dimensional calculation (i.e. without the dynamical effects of Ref. [^]) by about one
order of magnitude.
In Fig. 14 we compare our estimates for the spontaneous fission half-lives {Tgf) with the
experimental data and also with the results of dynamical calculations on the basis of the Saxon-
Woods potential made in Ref. ||2^. It is seen that we have qualitatively explained its decrease
for heavy Fm isotopes. It is due to the fact that the second hump goes below the ground state
level in 256-258pj^ fact, we did not reproduce well enough the half-life time for ^^^Fm which
is in our calculations 3 orders of magnitude shorter than the measured value and therefore, the
abrupt change in the systematics of the fission life-times appears 2 mass units too early.
4 Conclusions
In this paper we have discussed some properties of the potential energy surfaces of fissioning
even-even nuclei in the ^^^Fm region. All discussed nuclei exhibit two-hump fission barriers.
There is always one of the fission trajectories which leads to compact fission. Another path,
leading to an alternative mode corresponding to a spherical and an elongated fission fragment,
has been found for all these nuclei. The shape of the second hump of the potential barrier and
the relation between the two fission paths in the potential energy surface are crucial for the
way in which fission occurs in these nuclei.
In ^^^Fm fission follows only the elongated fission path but we are unable, in our static
calculation, to reproduce properly the mass asymmetry of the fission products of ^^^Fm. Sim-
ilarly to the experimental situation the theoretical approach yields only a low TKE mode in
the fission yields was found theoretically. In contrast to this, the isotope ^^^Fm may fission
along the CF or EF valleys and bimodal fission take place there. The observed experimental
difference of the TKE between both modes is well reproduced by our model. The mechanism
of bimodal fission of ^^^Fm and the heavier even-even nuclei has been described properly.
We also explained the decrease of the half-life times for the heavy Fm isotopes. In the case
of ^^"^Fm, the second hump on the fission barrier is located above the ground state. In heavier
isotopes it goes down by a few MeV below the ground state minimum and therefore does not
give any contribution to the half-life times of these nuclei.
Contrary to the majority of papers describing bimodal fission of ^^^Fm we have found a
strong left-right asymmetry in the shape of fissioning nucleus which nevertheless corresponds
to a symmetric mass split. This is a new phenomenon which could be discovered only in the
HFB type of calculations with the Gogny or 5-pairing forces which distinguish between orbitals
in different fragments.
Acknowledgments:
Two of the authors (K. P. and M. W.) gratefully acknowledge the warm hospitality extended
to them by the Departamento de Fi'sica Teorica of the Universidad Autonoma de Madrid as well
as grants from the Spanish Interministerial Commission of Science and Technology, Ref. SB99-
BA6182184 (M. W.) and of the Spanish Foreign Ministry (K. R).
This work is partly sponsored by DGICyT, Spain under Project PB97-0023 and the Polish
Committee of Scientific Research, grant No. 2P 03B 115 19.
The critical reading of the manuscript and the comments done by Johann Bartel is also
acknowledged.
11
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13
-1880
-1890
i^TZ 1
-1900
-1910
> -1920
'\ •
A
n
\
\\ •.
lij -1930
A/o = 13
"x ■•.
\\ '.
\\ *.
-1940
— A/o=15
*\ •.
A ■•
A ■•.
\\ •.
\\
^\
x\
-1950
■— A/o = 17
^\
^\
^\
^\ *•.
^\ •.
-1960
-1970
50 100 150 200 250
Figure 1: The HFB energy obtained for iVo=13, 15 and 17 for the compact fission (CP) path
of ^^^Fm as a function of the quadrupole moment Q^-
14
-1870 r--r
Q2[b\
Figure 2: The potential energy surface obtained with the HFB calculations (dotted line), the
one taking into account the two body energy correction (dashed line) and the one including also
the rotational energy correction (full line) for the compact fission path of ^^^Fm as a function
of the quadrupole moment Q2-
15
50 100 150
200 250
Figure 3: Panel (a): The fission barrier of ^^^Fm as a function of the quadrupole moment Q2
for A^o — 15. The sohd hne corresponds to the compact fission path (CF) and the dashed
hne to the elongated one (EF). The dotted line shows the reduction of the first barrier due
to nonaxial degrees of freedom. The shapes of the nucleus at a density of po =
are depicted for several values of Q2 both for the CF and EF paths (note that the EF path
leads to octupole deformed shapes). Additionally, in panels (b), (c) and (d) the octupole and
hexadecapole moment as well as the neck parameter Qn are respectively plotted.
16
200
150
f 100
50
Q2 = 200 ^ \ :
-20 -15 -10 -5 5 10 15 20
z[fm]
Figure 4: The shape of ^^^Fm at deformation Q2 = 200b on the elongated path to fission as well
as the number of particles (b) and the density (c) of protons (solid line) and neutrons (dashed
line) as a function of z. The number of particles corresponding to the magic numbers 50 and
82 are marked in the panel (b) with horizontal dotted lines.
17
15 -
10 :
5 :
-5 :
-10 :
-15 -
Figure 5: The cross section of the potential energy surface of ^^^Fm for different values of Q2
as a function of the neck parameter Qn-
18
60 I— r
-1980 r 1
50 100 150 200 250
Q^[b]
Figure 6: The same as in Fig. 3, but for the ^^^Fm nucleus.
19
-1890 -
>
^ -1900 -
S -1900 -
-1890 -
>
^ -1900 -
-1910 -
Figure 7: The same as in Fig. 5, but for the ^^^Fm nucleus.
20
-1030
10 15
Q^[b]
Figure 8: The potential energy surfaces for some Sn isotopes as a function of the quadrupole
moment Q2- Both the 2b-KEC and REC are included in the curves
21
22
Figure 10: The same as in Fig. 5, but for the ^^*No nucleus.
23
24
25
-30 '
50 100 150
Q2[b]
Figure 13: The fission barriers for ^^^Fm, ^^^Fm and ^^^Fm evaluated along the CF path. The
ground state is set to zero in the three cases. The second full hue starting at Q2 — 60 h
correspond to the EF path of ^^^Fm.
26
Figure 14: The spontaneous fission half-life times of Fm isotopes as a function of the mass num-
ber. The experimental data (exp.) are taken from the NuDat data base while the theoretical
estimates computed with a model based on the Woods-Saxon (WS) potential are taken from
Ref ■ ||25|| ■ The present estimates are represented by full circles for fission along the CF path
whereas the results for fission along the EF path are represented by open circles (note that
both estimates coincide in 256-258pj^^ these results were obtained assuming the axial symmetry
of fissioning nucleus. Similar estimates done with inclusion of the nonaxial degrees of freedom
are marked by triangles.
27