First, we have studied the stability of our results with respect
to the number of the deformed HO states
) used in this basis. Figure 1 shows the calculated
self-consistent potential energy curve of Pu as a
function of the axial quadrupole constraint
. The solid line corresponds to
=1000 deformed HO states included in
At each value of , the HO basis deformation is determined
as that corresponding to through the standard first-order
expression. Therefore, the set of HO states
included in the basis changes with the changing nuclear shape.
The results with different values of
(14 spherical oscillator shells included at =0), 816 (15), 969 (16), 1140 (17),
1330 (18), 1540 (19), and 1771 (20) are also shown at the extreme
points (minima and maxima) of the potential energy curve.
It is seen that the rate of convergence weakly depends on the quadrupole
deformation, and for larger elongations is slightly slower than at sphericity.
As shown in Fig. 1, reliable calculations can be carried
1140 (17), and the errors related to
the finite basis sizes that impact energies of
the first and second barrier are less than 1
Stability of the calculated SHF+BCS energy curve for
Pu as a function of the harmonic-oscillator basis size, see text.
Having determined the basis size, we performed the set of
calculations for the series of even-even Fermium isotopes which
are experimentally known to exhibit rapid variations of the
spontaneous fission half-lives.
Two fission paths corresponding to bimodal fission have been
investigated. The usual reflection-asymmetric path corresponds to two
fission fragments with different masses. The reflection-symmetric path
can be associated with division into symmetric, nearly spherical
fragments with high kinetic energies.
In our calculations, the quadrupole and octupole mass
moments were used as constraints.
octupole mass moment indicates the breaking of intrinsic
parity, while the presence of the quadrupole moment
signals the appearance of triaxial distortions.
and the mass
octupole moments calculated along the
fission paths for the even-even Fermium isotopes with
=142-152. The differences between the open and solid
symbols in the region of the first barrier represent the energy
gain due to triaxiality.
=1140 was used.
Deformation energy curves
) and the mass octupole moments () calculated
along static fission paths for the even-even Fermium isotopes
with the neutron numbers =142-164 are plotted in Figs. 2 and 3
of the mass quadrupole moment .
All calculated static barriers have similar, two-humped shapes.
The sizes of the static fission barriers are correlated
with the experimentally known increase in the
spontaneous fission half-lives for the lighter Fermium isotopes with
=142-152. The influence of triaxial asymmetry on the height of
the first fission barrier is shown as a difference between the open
and solid symbols. The effect of triaxiality
increases with neutron number and reaches about 3MeV for 150.
For b, the fission paths of nuclei displayed
in Fig. 2 exhibit reflection-symmetric shapes (), while the
reflection-asymmetric paths () takes over at larger
elongations with b.
For the heavier Fm isotopes displayed in Fig. 3, the transition
to the reflection-asymmetric path occurs at greater values
of and the transition point increases with . In the
extreme case of Fm (which is
expected to fission into two doubly magic Sn nuclei),
the fission path becomes reflection-symmetric.
The disappearance of the reflection-asymmetric path at large
neutron numbers gives rise to the reduction, or even disappearance,
of the second
fission barrier. This reduction plays a crucial
role in the standard interpretation of the experimentally known
rapid decrease of the spontaneous fission half-lives in the heavy Fm
Similar as in
Fig. 2 except for the even-even Fermium isotopes with =154-164.
In order to better understand the competition between
and reflection-asymmetric fission valleys, we show in Fig. 4
the total energy curves for
Fm calculated with a smaller
=680. For each nucleus
the two different fission paths are displayed: reflection-asymmetric
(open symbols) and reflection-symmetric (solid
symbols). It is seen that at large values of ,
the barriers along the reflection-asymmetric
paths are more narrow, whereas the distinctly higher second humps
calculated along the static fission path
for the even-even Fermium isotopes Fm. Open (closed) symbols
represent the reflection-asymmetric (symmetric) path. As compared
to Figs. 2 and 3,
the size of the oscillator basis is slightly reduced,
Figure 5 shows the deformation energy curves and the mass
hexadecapole moments calculated along the static fission paths of the
superheavy even-even nuclei with =184 (which is the neutron
magic gap in this region). These results
markedly differ from those obtained for the Fm isotopes. As
expected, all =184 isotones have spherical ground states.
The corresponding static two-humped fission barriers are narrow
The triaxial deformation lowers the first barrier, with
the maximum reduction (2 MeV) predicted for 124.
As illustrated in Fig. 5, with increasing , the height of
the first barrier increases
while the second barrier gets reduced. The increase in the hexadecapole moment along the fission
path reflects the development of the neck.
and the mass
hexadecapole moment as a function
of for the even-even superheavy nuclei with . The
differences between the open and solid symbols represent the effect
of triaxiality on the first barrier.