#  Adiabatic fission barriers in superheavy nuclei

P. Jachimowicz  Institute of Physics, University of Zielona Góra, Szafrana 4a,
65516 Zielona Góra, Poland  M. Kowal<EMAIL_ADDRESS>National Centre for
Nuclear Research, Hoża 69, PL-00-681 Warsaw, Poland  J. Skalski  National
Centre for Nuclear Research, Hoża 69, PL-00-681 Warsaw, Poland

###### Abstract

Using the microscopic-macroscopic model based on the deformed Woods-Saxon
single-particle potential and the Yukawa-plus-exponential macroscopic energy
we calculated static fission barriers $B_{f}$ for 1305 heavy and superheavy
nuclei $98\leq Z\leq 126$, including even - even, odd - even, even - odd and
odd - odd systems. For odd and odd-odd nuclei, adiabatic potential energy
surfaces were calculated by a minimization over configurations with one
blocked neutron or/and proton on a level from the 10-th below to the 10-th
above the Fermi level. The parameters of the model that have been fixed
previously by a fit to masses of even-even heavy nuclei were kept unchanged. A
search for saddle points has been performed by the ”Imaginary Water Flow”
method on a basic five-dimensional deformation grid, including triaxiality.
Two auxiliary grids were used for checking the effects of the mass asymmetry
and hexadecapole non-axiallity. The ground states were found by energy
minimization over configurations and deformations. We find that the non-
axiallity significantly changes first and second fission barrier in many
nuclei. The effect of the mass - asymmetry, known to lower the second, very
deformed barriers in actinides, in the heaviest nuclei appears at the less
deformed saddles in more than 100 nuclei. It happens for those saddles in
which the triaxiallity does not play any role, what suggests a decoupling
between effects of the mass-asymmetry and triaxiality. We studied also the
influence of the pairing interaction strength on the staggering of $B_{f}$ for
odd- and even-particle numbers. Finally, we provide a comparison of our
results with other theoretical fission barrier evaluations and with available
experimental estimates.

###### pacs:

25.70.Jj, 25.70.Gh, 25.85.Ca, 27.90.+b

##  I INTRODUCTION

Although fission barrier heights $B_{f}$ are not directly measurable
quantities, i.e. are not quantum observables, they are very useful in
estimating nuclear fission rates. As the activation energy $E_{a}$ (per mole)
in chemistry gives a rate $k$ of a chemical reaction at temperature $T$ via
the Arrhenius law: $k=Ae^{-E_{a}/RT}$ ($R$ \- the gas constant; $A$ \- the
frequency factor) tHoff ; Arrhe , the fission barrier gives the fission rate
$\Gamma_{f}$ of an excited (as they usually are in nuclear reactions) nucleus
via: $\Gamma_{f}\sim e^{-B_{f}/kT_{eff}}$, where $T_{eff}$ is an effective
temperature derived from the excitation energy, and $k$ \- the Boltzman
constant. For example, knowing fission barriers of possible fusion products
helps predicting a cross section for a production of a given evaporation
residue in a heavy ion reaction: one can figure out whether neutron or alpha
emission wins a competition with fission at each stage of the deexcitation of
a compound nucleus. Moreover, one can try to understand the experimentally
established, intriguing growth of the total cross sections around Z=118; for
its correlation with $B_{f}$, see e.g. Fig. 6 and the related discussion in O1
. On the other hand, the prediction of the spontaneous or low energy (i.e.
from a weakly excited state) fission rates, governed by the regime of the
collective quantum tunneling, requires an additional knowledge of the barrier
shape and mass parameters.

A non-observable status of the fission barrier, again in analogy to that of
the activation energy in chemistry, is reflected in its possible dependence on
a reaction type and/or the excitation energy (effective temperature) range.
This leads to some uncertainty in calculations of fission barriers. In
particular, it is not clear whether intrinsic configurations should be
conserved along the level crossings, which increases $B_{f}$, or the adiabatic
state should be followed. This is especially relevant for odd-$A$ and odd-odd
nuclei, in which sharp crossings of levels occupied by the odd particle
exclude the strictly adiabatic scenario. It is known that if the projection of
the single-particle angular momentum on the symmetry axis of a nucleus
$\Omega$ is conserved, the diabatic effect on the fission barrier can be huge,
see e.g. Kisomers . As there is no accepted formula for a barrier correction
due to the non-adiabaticity, it is usually ignored, even in odd-$N$ and/or
odd-$Z$ nuclei. A general idea is that at the excitation energies close to,
and higher than the barrier, but still not inducing sizable dissipative
corrections, the adiabatic barrier could be used for calculating fission
rates.

Since calculations of potential energy surfaces (PES’s) for odd-$A$ and odd-
odd nuclei involve a repetition of calculations for many low-lying
quasiparticle states which multiplies the effort (especially in odd-odd
systems), systematic studies of their fission barriers are rather scarce. Up
to now, they were provided mainly by the Los Alamos microscopic-macroscopic
(MM) model and recently by some self-consistent models Gorbar . The current
state of theoretical predictions in fission of even - even nuclei (with
Z$\geq$ 100) has been discussed recently in nucl2015 .

In the present paper we extend our MM model based on the deformed Woods-Saxon
potential, which up to now was applied mainly to even-even nuclei Kow , to
odd-$A$ and odd-odd SH systems. We study a wide range of isotopes which,
perhaps, may be of some use for astrophysical purposes. The fission barriers
are calculated using the adiabatic assumption, i.e. they are the smallest
possible. Since the model has been quite reasonable, in particular in
reproducing first Kow  and second IIbarriers  fission barriers in actinides,
as well as super- kowskal  and hyper-deformed IIIbarriers1 ; IIIbarriers2
minima, we prefer to keep its parameters unchanged. The shell and pairing
correction for an odd nucleon system is done by blocking the lowest-lying
quasiparticle states. The modification of the macroscopic energy by including
the average pairing energy contribution which we introduced for nuclear masses
in JachKowSkal2014  is irrelevant for fission barriers.

The other motivation of our study is to improve the predictions for the
fission saddles. This requires simultaneously taking into account a large
number of shape variables IIbarriers ; IIIbarriers2  and relying on an in
principle exact method for finding saddles to escape errors inherent in the
mostly used constrained minimization method, see Moller2009 ; Dubray . As
usual, to make the involved computational effort manageable one has to make
some compromises which will be discussed in detail. The need for a
simultaneous consideration of many shape variables in PES’s calculations is
common to all nuclear models, including self-consistent theories based on some
effective interactions Schunck . The results on fission saddles obtained up to
now in the SH region clearly show the great importance of triaxial
deformation, neglected in many published work. A recent study BroSkal  of
barriers within both the MM Woods-Saxon and Skyrme SLy6 Hartree-Fock plus BCS
models shows that triaxiality is even more crucial beyond $Z=$ 126.

A description of our method of calculations is given in section II. The
results, details of the additional calculations, and comparisons with other
calculated barriers are presented and discussed in section III. Finally, the
conclusions are summarized in section IV.

##  II The Method

Multidimensional energy landscapes are calculated within the MM model besed on
the deformed Woods-Saxon potential WS . The Strutinski shell and pairing
correction STRUT67  is taken for the microscopic part. For the macroscopic
part we used the Yukawa plus exponential model KN  with parameters specified
in MUNPATSOB . Thus, all parameter values are kept exactly the same as in all
recent applications of the model to heavy and superheavy nuclei

The main point in fission barrier calculations is its reliability which, once
the model for calculating energy of a nucleus as a function of deformation is
fixed, hangs on two main ingredients: 1) the kind and dimension of the
admitted deformation space and 2) a method applied to the search for saddles.

Mononuclear shapes can be parameterized via spherical harmonics ${\rm
Y}_{lm}(\vartheta,\varphi)$ (for brevity we will just use the symbol ${\rm
Y}_{\lambda\mu}$ ) by the following equation of the nuclear surface:

$R(\vartheta,\varphi)=c(\\{\beta\\})R_{0}\\{1+\sum_{\lambda=1}^{\infty}\sum_{\mu=-\lambda}^{+\lambda}\beta_{\lambda\mu}{\rm
Y}_{\lambda\mu}\\},$ (1)

where $c(\\{\beta\\})$ is the volume-fixing factor and $R_{0}$ is the radius
of a spherical nucleus. This parameterization has its limitations; certainly,
it is not suitable for too elongated shapes. However, for moderately deformed
saddle points in superheavy nuclei it excellently reproduces all shapes
generated by other parametrizations, e.g. by Mol2000 , as we checked in
numerous tests.

For nuclear ground states it is possible to confine analysis to axially-
symmetric shapes, with the expansion truncated at $\beta_{80}$:

$\displaystyle R(\vartheta,\varphi)$ $\displaystyle=$ $\displaystyle
c(\\{\beta\\})R_{0}\\{1+\beta_{20}{\rm Y}_{20}+\beta_{30}{\rm
Y}_{30}+\beta_{40}{\rm Y}_{40}$ $\displaystyle+$ $\displaystyle\beta_{50}{\rm
Y}_{50}+\beta_{60}{\rm Y}_{60}+\beta_{70}{\rm Y}_{70}+\beta_{80}{\rm
Y}_{80}\\}.$

Thus, a seven dimensional minimization is performed using the gradient method.
For odd systems, the additional minimization over configurations is performed
at every step of the gradient procedure. Considered configurations consist of
the odd particle occupying one of the levels close to the Fermi level and the
rest of the particles forming a paired BCS state on the remaining levels. Ten
states above and ten states below the Fermi level have been blocked and energy
minimized over these configurations.

The main problem in a search for saddle points is that, since they are neither
minima nor maxima, one has to know energy on a multidimensional grid of
deformations (the often used and much simpler method of minimization with
imposed constraints may produce invalid results Moller2009 ; Dubray ;
IIbarriers ; Schunck . To find saddles on a grid we used the Imaginary Water
Flow (IWF) technique. This conceptually simple and at the same time very
efficient (from a numerical point of view) method was widely used and
discussed before Luc91 ; Mam98 ; Hayes00 ; Moeler04 ; Moller2009 ; IIbarriers
. The number of numerically tractable deformation parameters
$\\{\beta_{\lambda\mu}\\}$ is practically limited. More than five-dimensional
grids, keeping in mind a subsequent interpolation, are intractable in
calculations for many ($\sim$ 1000) nuclei. Including mass- and axially-
symmetric deformations ($\beta_{20}$, $\beta_{40}$, $\beta_{60}$, $\beta_{80}$
\- see Patyk1 ; Patyk2 ; Patyk3 ; Smol1  together with both, mass-asymmetry
($\beta_{30}$, $\beta_{50}$, $\beta_{70}$) and triaxiality (at least
$\beta_{22}$) would mean at least an eight-dimensional mesh and was impossible
at present.

Based on our previous results showing that triaxial saddles are abundant in SH
nuclei Kow , we consider that quadrupole triaxial shapes have to be
necessarily included. We treated the effects of mass-asymmetry and nonaxial
higher multipoles as corrections and analysed them at the second stage of
calculations. A rationale for a lesser importance of mass-asymmetric saddles
is that, while they constitute a second, more deformed ($\beta_{20}\approx
0.7-0.8)$, prominent barrier peak in actinides, their heights are much reduced
in SH nuclei where they become irrelevant. In the remaining, less deformed
saddles the mass asymmetry occurs less frequently. As to the nonaxial
multipoles of higher order, they are less important for saddles with small to
moderate $\gamma$ [where $\gamma$ is the Bohr’s quadrupole nonaxiality
parameter, cf. Eq. (III.3)]. They become important for $\gamma$ closer to
$60^{o}$ where they are needed to produce oblate shapes having $x$ as the
symmetry axis. Thus, they should be included for nuclei with a large oblate
g.s. deformation and a short triaxial barrier. The additional studies of the
mass-asymmetry and higher nonaxial multipoles are described in the proper
subsections of the Results section.

Thus, at the first stage, for all 1305 investigated nuclei the saddle points
were searched in a five dimensional deformation space spanned by:
$\beta_{20}$, $\beta_{22}$, $\beta_{40}$, $\beta_{60}$, $\beta_{80}$, using
the IWF technique. The appropriate nuclear radius expansion has the form:

$\displaystyle R(\vartheta,\varphi)$ $\displaystyle=$ $\displaystyle
c(\\{\beta\\})R_{0}\\{1+\beta_{20}{\rm
Y}_{20}+\frac{\beta_{22}}{\sqrt{2}}\left[{{\rm Y}_{22}+{\rm Y}_{2-2}}\right]$
(3) $\displaystyle+$ $\displaystyle\beta_{40}{\rm Y}_{40}+\beta_{60}{\rm
Y}_{60}+\beta_{80}{\rm Y}_{80}\\}.$

The five-dimensional calculations are performed on the following deformation
mesh:

$\displaystyle\beta_{20}$ $\displaystyle=$ $\displaystyle\ \ 0.00\ (0.05)\
0.60$ $\displaystyle\beta_{22}$ $\displaystyle=$ $\displaystyle\ \ 0.00\
(0.05)\ 0.45$ $\displaystyle\beta_{40}$ $\displaystyle=$ $\displaystyle\ \
-0.20\ (0.02)\ 0.20$ $\displaystyle\beta_{60}$ $\displaystyle=$
$\displaystyle\ \ -0.10\ (0.02)\ 0.10$ $\displaystyle\beta_{80}$
$\displaystyle=$ $\displaystyle\ \ -0.10\ (0.02)\ 0.10$

This makes a grid of 29250 points which was subsequently interpolated to a
fivefold denser grid of 50735286 points with the step 0.01 in each dimension.
On the latter, the saddle point, or rather several saddle points - if there
were a few of comparable heights within the 0.5 MeV energy window - were
searched for by means of the IWF procedure. For odd or odd-odd nuclei, at each
grid point we were looking for low-lying configurations by blocking particles
on levels from the 10-th below to the 10-th above the Fermi level (in neutrons
or/and protons).

The fact that searches for ground states and for saddles are separated -
performed using different deformation spaces - allows saving some number of
deformation parameters in Eq. (3). This is equivalent to assuming that the
fission saddles have mostly prolate deformations large enough to make nonaxial
deformations of multipolarity $\lambda\geq 3$ less important. One has to check
this assumption afterwards and separately treat nuclei in which the inclusion
of nonaxial deformations with $\lambda\geq 4$ is necessary.

Although, as mentioned before, in SH nuclei the second barriers at large
deformations are usually smaller than the first one or do not exist at all,
for $Z=98$-101 the mesh (4) was extended to $\beta_{20}=1.5$ and the second
saddles were searched for by the IWF technique. It turned out that these more
deformed barriers are indeed mostly smaller than the first ones and decrease
with increasing $Z$. Only in Cf isotopes with $N=134-160$ there were some
second saddles (at $\beta_{20}\approx 0.9$) higher than the first one by at
most 0.5 MeV. However, even those saddles were lowered by at least 1 MeV after
including the mass-asymmetry. Therefore, we have reasons to believe that the
range of $\beta_{20}$ in (4) is sufficient for knowing the height of the
fission barrier in the whole studied region.

##  III Results and discussion

In the present paper we have systematically calculated fission-barrier heights
$B_{f}$ as the energy difference between the saddle point and the ground
state. The saddle point is defined as the minimum over possible fission paths
of the maximal energy along the path. Let us emphasize that the calculations
presented here have been performed without adding any zero-point vibration
energy. We have included 1305 heavy and superheavy nuclei with proton numbers
$98\leq Z\leq 126$ and neutron numbers in the range $134\leq N\leq 192$, with
the smallest $N$ for a given $Z$ increasing by one with every step in $Z$. All
obtained barriers have been collected in Table 3. On all PES’s presented here,
energy is normalized in such a way that its macroscopic part is set to zero at
the spherical shape.

###  III.1 Potential Energy surfaces

Some idea about the positions of ground states, secondary minima and saddles
may be gained from PES’s. Chosen examples are shown in figures for: 252Lr -
Fig. 1, 270Db - Fig. 2, 276Mt - Fig. 3, 280Cn - Fig. 4, and 297119 - Fig. 5.
Overall evolution of ground states with increasing $Z$ from prolate to
spherical can be seen there. In some nuclei one can see multiple saddles of
which the one defining the fission barrier should be properly chosen.
Sometimes the saddles between competing minima can be important, therefore the
determination of all saddles on the map is necessarily needed.

The energy landscapes Fig. 1-5 were obtained by minimizing energy on the 5D
grid (3) with respect to $\beta_{40}$, $\beta_{60}$ and $\beta_{80}$. One
should be aware of two related circumstances: 1) As the grid Eq. (3) does not
include nonaxial deformations $\lambda\geq 4$, the axial deformations
$\lambda=4,6,8$ with respect to the $x$-axis cannot be reproduced, so the
landscapes are inexact around the oblate $\gamma=60^{o}$ axis. 2) A reduction
of a $n$-dimensional grid of energy values via the minimization over $n-2$
deformations sometimes leads to an energy surface composed from disconnected
patches, corresponding to multiple minima in the auxiliary (those minimized
over) dimensions. This can distort the picture of the barrier (actually, a
reduction of multi-dimensional data to a two-dimensional map is a general
problem).

With these reservations in mind, one can still explore some of the details
shown in the maps. In particular, the prolate g.s. minimum with strongly
nonaxial first saddle point at $\beta_{20}=0.41$ and $\beta_{22}=0.18$ is
visible in 252Lr. One can notice that the axially symmetric saddle lies more
than 2 MeV higher.

Figure 1: Energy surface, $E-E_{mac}(sphere)$, for $Z=103$ and $N=149$.

.

A slightly less steep, prolate g.s. minimum and a gently emerging second
minimum is visible in Fig 2 for 270Db. The triaxial saddle at
$\beta_{20}=0.52$ and $\beta_{22}=0.13$ has a smaller triaxiality $\gamma$
than the saddle in 252Lr. A decrease in barrier height due to triaxiality is
$\approx$ 2 MeV, Fig. 2.

Figure 2: The same as in 1 but for $Z=105$ and $N=165$.

.

In a heavier nucleus 276Mt, a prolate deformation of the g.s. is clearly
smaller than in 252Lr, see Fig 3. The second minimum, which was barely
outlined in 270Db, is more pronounced here, giving the fission barrier a
double-hump structure. The deformation $\beta_{20}\approx 0.5$ of the second
saddle is much smaller than that of the second barriers in actinides. Thus, a
two-peak structure of the barrier in SH nuclei may be viewed as a result of a
division (split) of the first barrier, occurring with growing $Z$. The higher
second axial saddle is lowered by triaxiality by $\approx$ 1.5 MeV, but
eventually is still higher than the first axial saddle.

Figure 3: The same as in 1 but for $Z=109$ and $N=167$.

.

For 280Ds a topology of the PES is even more complicated. We see several
minima: prolate - the g.s. and a superdeformed one, and a shallow oblate. The
map shows also a few saddles. The axially deformed saddle point at
$\beta_{20}=0.3$ has a similar height as the nonaxial saddle at
$\beta_{20}=0.54$ and $\beta_{22}=0.12$. It follows from the IWF calculation
that the second fission barrier is nonaxial in this case. The axial second
saddle is lowered by $\approx 1$ MeV owing to the nonaxiallity.

Figure 4: The same as in 1 but for $Z=112$ and $N=168$.

.

Figure 5: The same as in 1 but for $Z=119$ and $N=178$.

.

The nucleus $Z=119$, $N=178$ is spherical in its g.s. - Fig. 5. There is a
secondary oblate minimum (whose depth is underestimated in the map due to
omission of nonaxial $\lambda=4,6$ deformations). There is a low triaxial
second saddle at $\beta_{20}\approx 0.5$ and two ”first saddles” with
different triaxiality, of which the one with a larger $\gamma$ is the fission
saddle.

Still another type of PES, typical of nuclei with the superdeformed oblate
g.s., is presented in Fig.8 in the subsection C.

###  III.2 Role of the mass asymmetry

To study the effect of the reflection (mass) - asymmetry on the fission
barriers, a two-step procedure has been performed. At the first stage, we have
checked the stability of all the saddles found on the basic 5D mesh (the
first, the second, …, axially symmetric or triaxial, of energy within 0.5 MeV
of the highest saddle) against the mass-asymmetry. This was done by a 3D
energy minimization with respect to $\beta_{30}$, $\beta_{50}$ and
$\beta_{70}$ around each saddle. Since most of the saddles are non-axial, the
most general version of our Woods-Saxon code had to be used. In this case,
when both symmetries (axial and mass symmetry) are broken simultaneously, the
nuclear shapes are defined by the following equation of the nuclear surface:

$\displaystyle R(\vartheta,\varphi)=R_{0}c(\\{\beta\\})\left\\{1\right.$
$\displaystyle+$ $\displaystyle\beta_{20}{\rm
Y}_{20}+\frac{\beta_{22}}{\sqrt{2}}\left[{{\rm Y}_{22}+{\rm Y}_{2-2}}\right]$
(5) $\displaystyle+$ $\displaystyle\beta_{30}{\rm Y}_{30}+\beta_{40}{\rm
Y}_{40}+\beta_{50}{\rm Y}_{50}$ $\displaystyle+$ $\displaystyle\beta_{60}{\rm
Y}_{60}+\beta_{70}{\rm Y}_{70}+\beta_{80}{\rm Y}_{80}\\}.$

It turned out that this minimization lowers energy of only those saddles in
which: i) there is no triaxiality, ii) deformation $\beta_{20}\approx 0.3$.
This supports an often expressed conventional ”wisdom”, that the mass-
asymmetry and triaxiality effects on fission saddle are decoupled. This is
why, at the second step of the procedure, we could carry out a full IWF
analysis on a grid including only axially-symmetric deformations:
$\beta_{20},\beta_{30},\beta_{40},\beta_{50},\beta_{60},\beta_{70},\beta_{80}$,
with $\beta_{20}$ restricted to a quite short interval $0.25-0.40$:

$\displaystyle\beta_{20}$ $\displaystyle=$ $\displaystyle\ \ 0.25\ (0.05)\
0.40$ $\displaystyle\beta_{30}$ $\displaystyle=$ $\displaystyle\ \ 0.00\
(0.05)\ 0.25$ $\displaystyle\beta_{40}$ $\displaystyle=$ $\displaystyle\ \
-0.15\ (0.05)\ 0.20$ $\displaystyle\beta_{50}$ $\displaystyle=$
$\displaystyle\ \ 0.00\ (0.05)\ 0.15$ $\displaystyle\beta_{60}$
$\displaystyle=$ $\displaystyle\ \ -0.10\ (0.05)\ 0.10$
$\displaystyle\beta_{70}$ $\displaystyle=$ $\displaystyle\ \ 0.00\ (0.05)\
0.15$ $\displaystyle\beta_{80}$ $\displaystyle=$ $\displaystyle\ \ -0.10\
(0.05)\ 0.10.$

This seven-dimensional grid, composed of 76800 deformations, was subject to
the fivefold interpolation in all directions before it was used in the IWF
procedure. This means that the IWF calculations have been performed on the
grid containing 1 690 730 496(!) points. We have made such 7-dimensional
analysis for more than 100 nuclei, for which the effect of minimization was
greater than 300 keV. Results for these nuclei are shown in Table 1. The rest
of 127 cases shown in Table 1 are the test nuclei, in which the effect of the
minimization was smaller than 0.3 MeV. The results for these additional nuclei
allow to appreciate whether the (in principle exact) IWF method could produce
a greater effect that the (inexact) minimization.

As one can see, the adopted procedure allowed to omit the problem of searching
for a saddle by using the (inexact) minimization method which is not always
reliable IIbarriers ; Moller2009 . For example, for Z=118 and N=165, the
discussed effect resulting from the minimization amounts to 0.44 MeV, which,
just in this case, is quite similar to 0.46 MeV obtained from the IWF
technique; however, in Z=113 and N=163 one obtains $\approx 0.5$ MeV
difference between saddles obtained by both methods. In this particular
nucleus, the $\approx 0.77$ MeV barrier lowering by the mass-asymmetry is the
largest among all studied nuclei. It should be also noted that for the
isotopes of Z = 113 the effect of the mass-asymmetry is particularly large,
see the the top panel in Fig. 6.

In the bottom panel of Fig. 6, we show the difference between the results of
the both methods - the minimization - (MIN) and ”Imaginary Water Flow” -
(IWF). One can see that this difference increases with the neutron number. In
particular, there is practically no effect derived from the mass-asymmetry in
${}^{281}113$ when IWF is used. On the contrary, the approach based on
minimization suggests still a quite substantial (spurious) effect (0.55 MeV).
One might notice that our conclusion concerning decoupling of the variables
describing the axial and reflection asymmetries is in a delicate contradiction
with the studies Lu .

Table 1:  Mass(reflection)-asymmetry effect on the fission barrier from the minimization - MIN and from the Imaginary Water Flow method - IWF (in MeV). | N | IWF | MIN | N | IWF | MIN | N | IWF | MIN
---|---|---|---|---|---|---|---|---|---
|  | $\mathbf{Z=109}$ |  |  | $\mathbf{Z=114}$ |  |  | $\mathbf{Z=117}$ |
| 157 | 0.39 | 0.81 | 155 | 0.28 | 0.59 | 157 | 0.24 | 0.34
| 158 | 0.22 | 0.42 | 156 | 0.14 |  $<$0.30 | 158 | 0.28 |  $<$0.30
| 159 | 0.54 | 0.45 | 157 | 0.72 | 0.83 | 159 | 0.24 | 0.34
| 160 | 0.31 | 0.54 | 158 | 0.46 | 0.46 | 160 | 0.12 |  $<$0.30
|  | $\mathbf{Z=110}$ |  | 159 | 0.67 | 0.68 | 161 | 0.26 |  $<$0.30
| 157 | 0.41 | 0.69 | 160 | 0.45 | 0.66 | 165 | 0.36 | 0.39
| 158 | 0.19 | 0.31 | 161 | 0.53 | 0.79 | 166 | 0.23 |  $<$0.30
| 159 | 0.52 | 0.46 | 162 | 0.42 | 0.64 | 167 | 0.19 | 0.50
| 160 | 0.50 | 0.40 | 163 | 0.58 | 0.65 | 168 | 0.07 |  $<$0.30
| 161 | 0.43 | 0.47 | 164 | 0.40 | 0.63 | 169 | 0.05 | 0.37
| 162 | 0.35 | 0.31 | 165 | 0.42 | 0.65 |  | $\mathbf{Z=118}$ |
|  | $\mathbf{Z=111}$ |  | 166 | 0.38 | 0.53 | 163 | 0.30 | 0.32
| 157 | 0.49 | 0.97 | 167 | 0.11 | 0.68 | 164 | 0.23 |  $<$0.30
| 158 | 0.36 | 0.78 | 168 | 0.06 | 0.41 | 165 | 0.46 | 0.44
| 159 | 0.61 | 0.83 |  | $\mathbf{Z=115}$ |  | 166 | 0.28 | 0.31
| 160 | 0.67 | 0.85 | 157 | 0.28 | 0.64 | 167 | 0.20 | 0.63
| 161 | 0.87 | 0.89 | 158 | 0.25 | 0.50 | 168 | 0.15 | 0.39
| 162 | 0.66 | 0.80 | 159 | 0.34 | 0.49 |  | $\mathbf{Z=119}$ |
| 163 | 0.56 | 0.83 | 160 | 0.39 | 0.38 | 165 | 0.46 | 0.57
| 164 | 0.58 | 0.68 | 161 | 0.56 | 0.58 | 166 | 0.33 | 0.37
| 166 | 0.48 | 0.49 | 162 | 0.42 | 0.39 | 167 | 0.34 | 0.49
|  | $\mathbf{Z=112}$ |  | 163 | 0.46 | 0.54 | 168 | 0.27 | 0.32
| 157 | 0.57 | 0.83 | 164 | 0.49 | 0.45 | 169 | 0.31 | 0.57
| 158 | 0.32 | 0.45 | 165 | 0.47 | 0.60 | 170 | 0.24 | 0.38
| 159 | 0.58 | 0.55 | 166 | 0.53 | 0.54 | 171 | 0.23 | 0.32
| 160 | 0.60 | 0.49 | 167 | 0.42 | 0.80 |  | $\mathbf{Z=120}$ |
| 161 | 0.51 | 0.60 | 168 | 0.20 | 0.55 | 165 | 0.39 | 0.38
| 162 | 0.53 | 0.48 | 169 | 0.13 | 0.31 | 166 | 0.17 |  $<$0.30
| 163 | 0.56 | 0.64 | 170 | 0.07 | 0.30 | 167 | 0.20 | 0.49
| 164 | 0.44 | 0.43 |  | $\mathbf{Z=116}$ |  | 168 | 0.15 |  $<$0.30
| 165 | 0.33 | 0.48 | 155 | 0.40 | 0.41 | 169 | 0.10 | 0.46
| 166 | 0.34 | 0.34 | 156 | 0.19 |  $<$0.30 |  | $\mathbf{Z=121}$ |
| 167 | 0.20 | 0.35 | 157 | 0.36 | 0.52 | 165 | 0.25 | 0.40
|  | $\mathbf{Z=113}$ |  | 158 | 0.26 | 0.34 | 166 | 0.23 |  $<$0.30
| 155 | 0.14 | 0.49 | 159 | 0.35 | 0.44 | 167 | 0.38 | 0.52
| 156 | 0.24 | 0.34 | 160 | 0.28 | 0.49 | 168 | 0.31 | 0.34
| 157 | 0.80 | 0.98 | 161 | 0.40 | 0.44 | 169 | 0.36 | 0.60
| 158 | 0.50 | 0.75 | 162 | 0.33 | 0.37 | 170 | 0.30 | 0.43
| 159 | 0.56 | 0.91 | 163 | 0.48 | 0.54 |  | $\mathbf{Z=122}$ |
| 160 | 0.61 | 0.88 | 164 | 0.40 | 0.38 | 164 | 0.00 |  $<$0.30
| 161 | 0.72 | 1.06 | 165 | 0.46 | 0.50 | 165 | 0.21 |  $<$0.30
| 162 | 0.57 | 0.93 | 166 | 0.33 | 0.40 | 166 | 0.12 |  $<$0.30
| 163 | 0.76 | 1.25 | 167 | 0.30 | 0.38 | 167 | 0.19 | 0.31
| 164 | 0.49 | 0.89 | 168 | 0.11 |  $<$0.30 | 168 | 0.11 |  $<$0.30
| 165 | 0.54 | 0.98 | 169 | 0.09 | 0.32 | 169 | 0.10 | 0.45
| 166 | 0.40 | 0.86 |  |  |  |  | $\mathbf{Z=123}$ |
| 167 | 0.19 | 0.78 |  |  |  | 166 | 0.06 |  $<$0.30
| 168 | 0.10 | 0.55 |  |  |  | 167 | 0.08 | 0.35
|  |  |  |  |  |  |  | $\mathbf{Z=124}$ |
|  |  |  |  |  |  | 165 | 0.23 | 0.31
|  |  |  |  |  |  | 166 | 0.06 |  $<$0.30
|  |  |  |  |  |  | 167 | 0.10 | 0.32
Figure 6:  Top panel: The fission barrier lowering by the mass-asymmetry
obtained by the (in principle exact) Imaginary Water Flow method - IWF and by
the (easier, but sometimes misleading) minimization method - MIN. Bottom
panel: The difference between both methods in MeV (in principle - the error in
the barrier height due to the minimization method).

###  III.3 Role of the triaxiality

The importance of including triaxiality in a calculation of fission barrier
heights was indicated many times before CWIOK92 ; CWIOK94 ; CWIOK96 ; GHERGH99
; DUTTA00 ; DECHARGE2003 ; BONNEAU04 ; Cwiok05 ; DOB2007 ; KOWAL2009 . In
particular, it was shown that the effect of both quadrupole and a general
hexadecapole nonaxiality, when accounted for within the nonexact method of
constrained minimization (used generally in all selfconsistent studies), may
reach 2.5 MeV for some superheavy even-even nuclei, see Fig. 5 in Kow . Here,
we extend our previous discussion of its role to the odd and odd-odd nuclei
and, at the same time, improve the treatment by employing the exact IWF method
in potentially most interesting cases.

By using the original 5D mesh (4) we have obtained saddles with quadrupole
nonaxiality for about 900 nuclei, what constitutes more than 70 % of all
fission barriers. We illustrate this conspicuous effect in Fig. 7 on the
example of two isotopic chains, Z=103 and 113.

Figure 7:  Effect of the non-axiallity on the fission barrier heights (see
text for further explanations).

We show the difference between axial and nonaxial barriers in these nuclei.
One can see that for lighter Lawrencium isotopes the effect of nonaxiality is
quite considerable. Starting with $N=$164, it is weakening quickly and finally
vanishes for $N\geq$ 176\. Somewhat different dependence of the effect on the
neutron number occurs in $Z=113$ isotopes. The maximum lowering of the barrier
of more than 1.5 MeV occurs for $N\approx 165$, there is a second maximum at
$N=179$, and the effect becomes large again at $N=192$. Inbetween, for
$N\approx 154$ and $N\approx 174$, there is no effect at all. Thus, the effect
of nonaxiallity has to be studied carefully, indeed.

Another task is to consider the influence of the hexadecapole nonaxiality,
namely: $\beta_{42},\beta_{44}$ in Eq. 1, on the fission barriers. The
unconstrained inclusion of these shapes would lead to a 7D grid which is too
much for now. To evaluate the effect without increasing the grid dimension we
constrained $\beta_{42}$ and $\beta_{44}$ to be functions of the quadrupole
nonaxial deformation $\beta_{22}$, or actually $\gamma$, and $\beta_{40}$, in
a well known manner geom . Using the conventional notation:

$\displaystyle\beta$ $\displaystyle=$
$\displaystyle\sqrt{\beta_{20}^{2}+\beta_{22}^{2}},$ $\displaystyle\gamma$
$\displaystyle=$ $\displaystyle{\rm arctg}\frac{\beta_{22}}{\beta_{20}},$ (7)

the following form of Eq. 1 was used:

$\displaystyle R(\vartheta,\varphi)=c({\beta})R_{0}\left\\{1\right.$
$\displaystyle+$ $\displaystyle\beta\cos{(\gamma)}{\rm Y}_{20}$ (8)
$\displaystyle+$ $\displaystyle\frac{\beta\sin{(\gamma)}}{\sqrt{2}}\left[{{\rm
Y}_{22}+{\rm Y}_{2-2}}\right]$ $\displaystyle+$
$\displaystyle\beta_{40}\frac{1}{6}(5\cos^{2}{(\gamma)}+1){\rm Y}_{40}$
$\displaystyle-$
$\displaystyle\beta_{40}\frac{1}{6}\sqrt{\frac{15}{2}}\sin{(2\gamma)}\left[{{\rm
Y}_{42}+{\rm Y}_{4-2}}\right]$ $\displaystyle+$
$\displaystyle\beta_{40}\frac{1}{6}\sqrt{\frac{35}{2}}\sin^{2}{(\gamma)}\left[{{\rm
Y}_{44}+{\rm Y}_{4-4}}\right]$ $\displaystyle+$ $\displaystyle\beta_{60}{\rm
Y}_{60}+\beta_{80}{\rm Y}_{80}\\}.$

On this 5D grid, the hexadecapole nonaxiality (but not the $\beta_{60}$ and
$\beta_{80}$ terms) preserves the modulo-60o invariance in $\gamma$, so, in
particular, the parameter $\beta_{40}$ describes a deformation which is
axially symmetric around the $z$ axis at $\gamma=0^{o}$ and around the $x$
axis at $\gamma=60^{o}$, which allows to better approximate energy at oblate
shapes. For this reason, while the original mesh Eq. (3) may be expected more
reliable for barriers at small $\gamma$, the one of Eq. (8) is better for
saddles closer to $\gamma=60^{o}$, like those in nuclei with well- or super-
deformed oblate ground states.

Our method of proceeding is analogous to that used in the study of the mass-
asymmetry. The difference is that we do not have to perform the first step: a
minimization with respect to $\beta_{42}$ and $\beta_{44}$ at the saddles
found from the grid Eq. (3). Such calculations were already done in the
previous studies of the effect of nonaxial deformations of higher
multipolarity on the fission barrier in heaviest nuclei KowSob1 ; KowSob2 ;
KowSob3 ; SobJachKow . We know that the minimization gave the largest effect
in the following four regions of nuclei, see Fig. 2 in SobJachKow : (I)
$Z\approx 122$, $N\approx 160$ \- up to 1.5 MeV, and a $\sim$ 3 times smaller
effect for nuclei with larger $N$ and $Z>120$, (II) $Z\approx 110$, $N\approx
146$ \- up to 1 MeV, (III) $Z\approx 114$, $N\approx 184$ \- up to 1 MeV, and
(IV) $Z\approx 104$, $N\approx 170$ \- up to 0.4 MeV.

By applying the IWF method on the mesh Eq. (8) we have found the saddles for a
dozen of nuclei from the last three regions, for which the effect of
minimization was the largest. It turned out that, compared to saddles found on
the original grid Eq. (3), they were lowered by less than 150 keV in the
region (II), by less then 100 keV in the region (III), and even increased by
$\sim$ 100 keV in the region (IV). On this basis we conclude that the lowering
of the fission saddles found by the minimization in Kow ; KowSob3  in these
three regions is in a large measure a spurious effect which mostly vanishes
when saddles are fixed by a proper method.

On the contrary, the substantial effect (up to $\approx 1$ MeV) of the
nonaxial hexadecapole in the region (I), although smaller than found by the
minimization, survives in the exact IWF treatment. This might be expected as
these are very heavy $Z\geq 119$ nuclei with short barriers and oblate (also
superdeformed) ground states, so $\beta_{42}$ and $\beta_{44}$ are necessary
to reproduce energy in the vicinity of the oblate axis. Therefore, in the
whole region of nuclei with $Z\geq 118$ we calculated triaxial barriers by the
IWF method using the mesh Eq. (8) and then selected the proper fission
barriers from two 5D calculations.

Three types of saddles in nuclei from the region (I) are shown for a very
heavy and exotic nucleus ${}^{285}122$ in Fig. 8. The landscape was created
from the 5D mesh Eq. (8).

Figure 8: Energy surface, $E-E_{mac}(sphere)$, for the nucleus $Z=122$,
$N=163$, resulting from the calculation according to Eq. (8).

This nucleus has a global superdeformed oblate (SDO) minimum with the
quadrupole deformation $\beta_{20}=-0.455$ (spheroid with the axis ratio
$\approx$ 3:2). It represents a neutron-deficient area of superheavy nuclei
according to recent predictions SDO . These intriguing SDO minima were already
confirmed, as the global ones, by various self-consistent models staszproch ;
skal2015 . There is a saddle close to the oblate axis, separating the SDO g.s.
from the wide minimum near the spherical shape - type a); the axially
symmetric saddle is designated as b). One fission path may go through the
saddles a) and b), the higher of which would define the barrier along this
path. The second fission path goes through a triaxial saddle of type c) at
$\beta_{20}\approx 0.4$, $\gamma\approx 35^{o}$. The fission barrier of
$B_{f}=3.6$ MeV corresponds to the saddle c) as found by using the grid Eq.
(8). It turns out that saddles of type a) and c) are much lowered by including
$\beta_{42}$, $\beta_{44}$, the first usually more than the second.

Table 2 summarizes the effect of nonaxial hexadecapole on the barriers in the
region (I). It contains 75 nuclei in which the barrier lowering is greater
than 300 keV. The most frequent saddle type in the region (I), on both grids,
is c), but there are also more complicated cases in which the saddle type
changes when $\beta_{42}$ and $\beta_{44}$ are included. The largest effect of
1.167 MeV occurs in the nucleus $Z=125$, $N=163$.

Let us remark that the difference between the results of the constrained
minimization and the IWF method for the nonaxial hexadecapole is the main
source of the discrepancy between the current fission barriers and those
published in Kow  for even-even nuclei.

Table 2:  The barrier lowering (in MeV) greater than 0.3 MeV in nuclei $Z\geq 118$, in particular in those with SDO ground states, from the IWF calculations on the 5D mesh including $\beta_{42}$ and $\beta_{44}$ according to Eq. (8). Also reported is the associated change in the saddle type (for a description of saddle types see text); no entry means that a c-type saddle results from both grids, Eq. (3) and (8). | N | $\Delta B_{f}$ | saddle | N | $\Delta B_{f}$ | saddle | N | $\Delta B_{f}$ | saddle
---|---|---|---|---|---|---|---|---|---
|  | $\mathbf{Z=119}$ |  |  | $\mathbf{Z=122}$ |  |  | $\mathbf{Z=125}$ |
| 155 | 0.597 |  | 158 | 0.779 |  | 161 | 1.083 |
| 156 | 0.482 |  | 159 | 0.959 |  | 162 | 0.958 |
| 157 | 0.472 |  | 160 | 0.807 |  | 163 | 1.167 |
| 158 | 0.566 |  | 161 | 0.731 |  | 164 | 0.936 |
| 159 | 0.585 |  | 162 | 0.690 |  | 165 | 0.439 | a$\rightarrow$c
| 160 | 0.508 |  | 163 | 0.469 | a$\rightarrow$c | 166 | 0.806 | b$\rightarrow$c
| 161 | 0.315 | b$\rightarrow$c | 164 | 0.364 | b$\rightarrow$c | 167 | 0.806 |
| 162 | 0.471 | a$\rightarrow$c | 169 | 0.403 | b$\rightarrow$c | 168 | 0.800 |
| 170 | 0.343 | b$\rightarrow$c | 170 | 0.365 |  | 169 | 0.714 |
| 172 | 0.480 | b$\rightarrow$c |  | $\mathbf{Z=123}$ |  | 170 | 0.551 |
| 173 | 0.501 |  | 159 | 0.831 |  |  | $\mathbf{Z=126}$ |
| 174 | 0.400 |  | 160 | 0.821 |  | 162 | 0.995 |
|  | $\mathbf{Z=120}$ |  | 161 | 0.863 |  | 163 | 1.099 |
| 156 | 0.613 |  | 162 | 0.924 |  | 164 | 1.034 |
| 157 | 0.731 |  | 163 | 0.496 | a$\rightarrow$c | 165 | 0.802 |
| 158 | 0.652 |  | 164 | 0.480 | a$\rightarrow$c | 166 | 0.912 |
| 159 | 0.778 |  | 168 | 0.357 | b$\rightarrow$c | 167 | 0.807 |
| 160 | 0.696 |  | 169 | 0.300 | b$\rightarrow$c | 168 | 0.845 |
| 161 | 0.658 |  |  | $\mathbf{Z=124}$ |  | 169 | 0.911 |
| 162 | 0.581 | a$\rightarrow$c | 160 | 0.819 |  | 170 | 0.735 |
| 163 | 0.323 | a$\rightarrow$b | 161 | 0.868 |  | 171 | 0.534 |
|  | $\mathbf{Z=121}$ |  | 162 | 0.896 |  | 172 | 0.434 |
| 157 | 0.747 |  | 163 | 0.741 |  |  |  |
| 158 | 0.774 |  | 164 | 0.739 | a$\rightarrow$c |  |  |
| 159 | 0.690 |  | 165 | 0.333 | b$\rightarrow$c |  |  |
| 160 | 0.830 |  | 166 | 0.334 | b$\rightarrow$c |  |  |
| 161 | 0.688 |  | 167 | 0.455 | b$\rightarrow$c |  |  |
| 162 | 0.633 | b$\rightarrow$c | 168 | 0.519 | b$\rightarrow$c |  |  |
|  |  |  | 169 | 0.459 |  |  |  |
|  |  |  | 170 | 0.328 |  |  |  |

###  III.4 Isotopic dependence

Calculated fission barriers given in Table 3 are illustrated along isotopic
chains in Figures: 9 \- 18. Generally, it can be seen that: i) in the whole
region $Z=$98 - 126 the fission barrier heights are limited by: $B_{f}\leq
8.06$ MeV; ii) there are characteristic maxima of fission barriers at
$Z\approx 100$, $N\approx 150$, near $Z=108$, $N=162$ (deformed magic shells)
and $Z=114$, $N=178$ (not 184); high barriers occur also at the border of the
studied region, for $Z=98$, $N\approx 183$; iii) over intervals of $N$ where
$B_{f}(N)$ increase or are on average constant, the fission barriers in a
neighboring system $N_{even}+1$ are higher than $B_{f}(N_{even})$; it may the
opposite over intervals where $B_{f}(N)$ strongly decrease; the same behaviour
can be seen when comparing barriers for isotones - see Fig. 20. This quite
pronounced odd-even staggering in barriers is related to a decrease in the
pairing gap due to blocking as it will be discussed in the next subsection.

In the isotopic dependence of the fission barriers for Cf, Es and Fm nuclei,
shown in Fig. 9, there are two peaks of a similar size, at $N=$ 152 and $N=$
184\. The minima of $B_{f}(N)$ occur at $N\approx 170$. Odd-even staggering in
$B_{f}$ for Es is stronger around $N=152$, while for Cf it is stronger near
$N=184$.

Figure 9: Isotopic dependance of fission barriers for $Z=98,99$ and $Z=100$.
Figure 10: The same as in Fig.9 but for $Z=101,102$ and $Z=103$. Figure 11:
The same as in Fig.9 but for $Z=104,105$ and $Z=106$. Figure 12: The same as
in Fig.9 but for $Z=107,108$ and $Z=109$. Figure 13: The same as in Fig.9 but
for $Z=110,111$ and $Z=112$. Figure 14: The same as in Fig.9 but for
$Z=113,1143$ and $Z=115$. Figure 15: The same as in Fig.9 but for $Z=116,117$
and $Z=118$. Figure 16: The same as in Fig.9 but for $Z=119,120$ and $Z=121$.
Figure 17: The same as in Fig.9 but for $Z=122,123$ and $Z=124$. Figure 18:
The same as in Fig.9 but for $Z=125$ and $Z=126$.

For Md, No and Lr isotopes (Fig. 10), the second maximum around $N=$184 is
weakening. A maximum associated with the semi-magic deformed shell at $N=$162
appears. As before, the minima of $B_{f}(N)$ are located at $N\approx 170$.
For Rf, Db, Sg, Bh and Hs nuclei (Fig. 11 and 12), previously distinct maximum
at $N=$152 becomes more flat, and a kind of plateau forms between $N=$152 and
162. For Mt isotopes this plateau changes into a local minimum in the isotopic
dependence $B_{f}(N)$, located around $N=$155\. The highest barriers in Bh, Hs
and Mt isotopic sets occur at $N\approx 162$.

For Ds, Rg and Cn nuclei (Fig. 13), with the increasing proton number the
$N=$184 spherical shell starts to dominate. However, not much lower barriers
are obtained near the deformed gap $N=$162.

For nuclei: $Z=$ Nh, Fl, Mc (Fig. 14), one can see one region with high
barriers, around $N=$180\. One can notice that the maxima in $B_{f}(N)$ are
already shifted toward $N<184$. Slight residues of the formerly observed
shells at $N=$152 and $N=$162 can be spotted.

For nuclei: $Z=$ Lv, Ts, Og (Fig. 15), the main maximum in $B_{f}$ progresses
further towards smaller $N$, reaching finally $N\approx 175$. The minima in
$B_{f}(N)$, observed before at $N=172$, gradually disappear. For nuclei:
$Z=$119, Z=120, Z=121 (Fig. 16), the situation is similar to that described
above. Barriers in nuclei $Z=$122, 123, 124 (Fig. 17), compared to the
previous set, are clearly lower. The maximum is even more shifted towards
smaller $N$. For nuclei: $Z=$125, 126 (Fig. 18) the fission barriers are still
lower. Their maxima occur at $N=171$ and 173.

All calculated fission barriers heights are collected together and shown as a
map $B_{f}(Z,N)$ in Fig. 19. One can see three areas with clearly raised
barriers: around $N\approx$152, $N=$162 and $N\approx 180$, and the region of
low barriers around $N=170$, as discussed above. The effect of the odd
particle, i.e. an often (but not always) higher barrier in a neighboring odd-
particle system can be also seen in Fig. 19.

Figure 19:  Calculated fission barrier heights $B_{f}$ for superheavy nuclei.

###  III.5 Role of the pairing interaction and the odd-even barrier staggering

It is known that the blocking procedure often causes an excessive reduction of
the pairing gap in systems with an odd particle number. This effect is much
more pronounced in the g.s. than in the fission saddle, as the pairing gap is
never small in the latter. One device to avoid an excessive even-odd
staggering in nuclear binding was to assume a stronger (typically by $\sim$
5%) pairing interaction for odd-particle-number systems, see Gor0 ; Gor1 ;
Gor2 ; Gor3 . Here, instead of performing another grid calculation with
modified pairing strengths, we tested the magnitude of their effect on fission
barriers by increasing them by 5 and 10 percent for odd particle numbers
(neutrons or protons) at previously found ground states and saddle points. The
results of this test are presented in Fig. 20 for the N=169 isotones and in
Fig. 21 for the Z=109 isotopic chain.

Both the isotopic and isotonic dependence show that increasing the intensity
of pairing leads to a reduction of the fission barrier by a variable amount.
When the pairing strengths are increased by 5% for odd particle numbers, the
fission barriers decrease in odd-even, even-odd and odd-odd systems by up to
0.5 MeV; the 10% increase in the pairing strengths can decrease the barriers
at most by about 1 MeV. The same pairing change leads to the suppression, and
then the inversion of the staggering effect.

The even-odd barrier staggering related to pairing is convoluted with the
isotopic or isotonic dependence related to the mean-field. With the original
pairing, when one separates a linear part of the latter by calculating:
$B_{f}(Z_{odd},N)-1/2[B_{f}(Z_{odd}+1,N)+B_{f}(Z_{odd}-1,N)]$, and an
analogous quantity for odd neutron numbers, one obtains numbers between 1.053
and $-0.947$ MeV, with the average of $\approx 0.22$ MeV for protons and
$\approx 0.26$ MeV for neutrons. As shown by black points in Fig. 20, 21, the
effect is indeed irregular and, when present, typically at the level of
several hundred keV.

Figure 20:  Effect of the pairing strength increase (while keeping fixed the
g.s. and saddle deformations) in N=169 isotones: standard $G_{n}$ and $G_{p}$
\- black points, $G_{n}$ and $G_{p}$ increased by 5% (10%) for odd-$Z$ and
odd-$N$ nuclei - red (green) points.  Figure 21:  Effect of the pairing
strength increase (while keeping fixed the g.s. and saddle deformations) in
Z=109 isotopes: standard $G_{n}$ and $G_{p}$ \- black points, $G_{n}$ and
$G_{p}$ increased by 5% (10%) for odd-$Z$ and odd-$N$ nuclei - red (green)
points.

The 5% increase in pairing for odd particle numbers reduces the staggering in
$N=169$ isotones and nearly cancels it in $Z=109$ isotopes (red points in
Fig.20 and 21). The important point is that the 10% increase in pairing for
odd number of particles inverts the staggering, at least locally: near $Z=120$
in $N=169$ isotones and near $N=153$, $N=162$ and $N=180$ in Mt isotopes
(green points in Fig.20 and 21).

Although the spontaneous fission rates of odd-particle number nuclei are
smaller by 3-5 orders of magnitude than those of their even neighbors, the
experimental fission barriers in actinides show only a moderate odd-even
staggering, c.f. SMIR93 ; OBN . Still, it is inconceivable that the fission
barriers in odd-$Z$ or odd-$N$ systems should be on average smaller than in
their even neighbors. This indicates that the 10% increase in pairing
strengths in odd-$N$ or odd-$Z$ systems would be too large. A qualitative
argument which follows is that even if the blocking method overestimates the
pairing decrease, the fission barriers of odd-$Z$ or/and odd-$N$ nuclei should
fall in a strip between the black and red points in Fig.20 and 21. Thus, the
test of the pairing influence on barriers points that a possible overestimate
of barriers in odd-$A$ and odd-odd nuclei, induced by the blocking, should not
be much larger than 0.5 MeV. One may add in this context that the barriers
from the FRLDM model do not show any even-odd staggering due to the way the
pairing was included there.

###  III.6 Comparison with other theoretical calculations and some empirical
data

Let us discuss the results in Table 3 in relation to available empirical data
and to the other theoretical estimates.

As an empirical check of our model, one can use the barriers in the actinide
region. We have reported quite a spectacular agreement of the calculated first
Kow  and second IIbarriers  fission barriers in even-even actinides with the
data SMIR93 ; OBN , with root mean square deviation 0.5 MeV and 0.7 MeV,
respectively.

The heaviest nucleus in which the fission barrier height has been measured
recently is 254No. The value $B_{f}$=6.0 $\pm$ 0.5 MeV at spin 15$\hbar$,
giving by extrapolation $B_{f}$=6.6 $\pm$ 0.9 MeV at the spin 0$\hbar$, has
been deduced from the measured distribution of entry points in the excitation
energy vs. angular momentum plane seweryniak . This result perfectly agrees
with our evaluation: $B_{f}$=6.88 MeV (at spin 0$\hbar$ ) and with the MM
model FRLDM  which gives: $B_{f}$=6.76 MeV. The selfconsistent calculations,
mainly based on the Skyrme interaction, overestimate this barrier
significantly Bonneau ; Duguet ; staszczak2006  (9.6 and 8.6 or 12.5 MeV,
respectively). There are experimental estimates of barriers in a few SH
nuclei, based on observed ER production probabilities ITKIS , which again well
agree with our barriers, see Kow . Apart from those, fission barriers in the
SH region are generally unknown.

As a supplementary insight, one can crosscheck barriers evaluated within
various models. Quite recently we noted a dramatic divergence in calculated
fission barriers japan2015 . Since, as it was discussed previously, the
inclusion of traxiality is absolutely necessary in the SH region, we have
chosen only models which take this into account. In fact, there is only one
systematic calculation, including triaxiality and odd-particle-number nuclei -
the Finite Range Liquid Drop Model mol95 ; Moller2009 ; FRLDM  (FRLDM)
developed by Los Alamos group. It can be noted though, that the inner fission
barrier is fixed there in only three-dimensional deformation space, what is
certainly not enough.

The first conclusion from the comparison between our results and those of
FRLDM is that a conspicuous barrier staggering between odd- and even-particle
number nuclei is obtained in the Woods-Saxon model. As mentioned before, this
results from the blocking treatment of pairing. At present it is not certain
how large this staggering should be.

One can include more models for comparison if one confines it to even - even
nuclei. We take the covariant density functional model RMF  with the nonlinear
meson-nucleon coupling, represented by the NL3* parametrization of the
relativistic mean-field (RMF) Lagrangian and the Hartree-Fock-Bogoliubov (HFB)
approach with the SkM* Skyrme energy density functional SKM .

Figure 22:  Fission barriers predicted by various models for Hassium isotopes:
black - WS model, green – FRLDM FRLDM , blue – SkM* SKM , red – RMF with NL3
parametrization RMF . Experimental data taken from ITKIS . (For interpretation
of the references to color in this figure legend, the reader is referred to
the web version of this article.)

As can be seen in Fig. 22, fission barriers in Hassium nuclei are quite
similar in all models. The values of $B_{f}$ differ up to 2 MeV, but never
more. Regarded as a function of $N$, they show a maximum close to the semi-
magic number $N=$162 while the second maximum is related with the $N=$184
spherical gap. In the FRLDM this maximum is barely outlined and slightly
shifted to the neutron deficient side. The minimum in barriers is obtained in
both MM models at the similar place ($N=$170), while the RMF gives the
smallest barriers at $Z=$174.

Figure 23:  The same as in Fig 22 but for Z=114.

As one can see in Fig. 23, for Flerovium isotopes the barriers calculated here
are in agreement with the experimental (empirical) estimates ITKIS  and with
the self-consistent calculations SKM  based on the SKM* interaction. The FRLDM
FRLDM  overestimates these quasi-empirical barriers ITKIS  significantly.
Although only the lower limit for the barrier height has been estimated in
ITKIS , which would reproduce the known cross sections on the picobarn level,
such a high barrier seems problematic, see discussion in Wilczynska1 ;
Wilczynska2 . On the other hand, with extremely small barriers obtained within
the RMF model one cannot explain experimentally known millisecond fission
half-life in 284Fl. One should note, however, that a slight tuning of the RMF
model RMF2015  gives higher barriers, thus, closer to ours. This is true,
especially in Cn and Fl isotopes, see details in Fig. 5 in RMF2015  and
discussion included there.

Figure 24:  The same as in Fig 22 but for Z=120.

For $Z=$120 our results, shown in Fig. 24, are very close to those obtained
within the RMF model. The results of FRLDM  are systematically higher by
$\approx$ 1 MeV. This is in an evident contrast with the Skyrme SkM*
prediction SKM  of the highest barriers for $Z=120$ SKM  \- related to the
proton magic gap. Three models: FRLDM, RMF and ours converge at N=182-184 to
$B_{f}\simeq 5$ MeV. The nucleus 302120 is particulary interesting, as two
unsuccessful attempts to produce it have already taken place in GSI, providing
a cross-section limit of 560 fb 120GSI1  or 90 fb in 120GSI2 , and in Dubna
120DUBNA , providing the limit of 400 fb. The cross-section estimates
Wilczynska3  do not support a possibility of an easy production of this SH
isotope in the laboratory. It seems that with the barrier of the order of 10
MeV, as obtained in the frame of the self-consistent theory, producing
superheavy Z=120 nuclei should not pose any difficulties.

Figure 25:  The same as in Fig 22 but for Z=126.

In the case of $Z=$126, shown in Fig. 25, both MM models give significantly
smaller barriers than the model based on the SKM* force. For example, the
barrier $B_{f}\approx 9$ MeV for ${}^{310}126$, calculated with this Skyrme
interaction, is still impressively large. This might induce thoughts on the
ways of synthesis of such superheavy systems, but one has to remember that the
predicted half-lives with respect to $\alpha$ decay are below the present-day
$10^{-5}$ s time-limit for the experimental identification. On the contrary,
$B_{f}\approx 2$ MeV obtained in the MM approach does not induce any hopes; it
only points to a quite striking disagreement between models.

##  IV Conclusions

We have determined fission barriers for 1305 heavy and superheavy nuclei,
including odd-$A$ and odd-odd systems, within the macroscopic-microscopic
method by following the adiabatic configuration in each nucleus. The applied
Woods-Saxon model was widely used for heavy nuclei and well reproduces
experimental fission barriers in actinides. For odd-$Z$ or/and odd-$N$ nuclei
pairing was included within the blocking procedure. Triaxial and mass-
asymmetric deformations were included and the IWF method used for finding the
saddles which allowed to escape errors inherent in the constrained
minimization approach. To find saddles, energy for each nucleus was calculated
on a 5D deformation grid and then 5-fold interpolated in each dimension for
the IWF search. Two additional energy grids: a second 5D and another 7D, were
calculated in order to include nonaxial hexadecapole and mass-asymmetry
effects on fission barriers. The following conclusions can be drawn from our
investigation:

i) Global calculations confirm the existence of two physically important areas
in the $Z$-$N$ plane with prominent barriers: one located around the semi-
magic quantum numbers $Z=100-108$ and $N=150-162$ (connected with deformed
closed shells) and the second - of nearly spherical nuclei around $Z=114$ and
$N=176-180$. The highest fission barrier among the studied nuclei occurs in
very exotic Es250.

ii) The well-known effect of the mass asymmetry on the second barrier in
actinides is not very relevant for the heaviest nuclei since very deformed
barriers at $\beta_{20}\approx 0.8$ decrease with increasing $Z$ and fission
barriers are fixed by the less deformed saddles. However, in some nuclei with
$Z\geq 109$ the mass(reflection) asymmetry effect lowers the first saddles
which are sometimes split into two humps. It seems that this concerns only
axially-symmetric saddles. The largest barrier lowering (by 0.8 MeV) has been
observed for $Z=113$ and $N=157$.

iii) It has been demonstrated that the inclusion of triaxial shapes
significantly reduces the fission barriers by up to 2.5 MeV; about 70% of the
found fission barriers correspond to triaxial saddles. Besides the quadrupole
nonaxiality we checked also the effect of hexadecapole nonaxiality which
significantly lowers the fission barrier in $Z\geq 119$ nuclei, especially
neutron-deficient ones.

iv) Rather strong, irregular odd-even $Z$ or $N$ barrier staggering effect
resulted from the blocking formalism used for pairing. The barrier of an odd
nucleus $Z_{even}+1$ or $N_{even}+1$ is typically by several hundred keV
higher than that of its even neighbor.

v) The existing theoretical evaluations of fission barriers differ
significantly. Even the results of the two models based on the microscopic-
macroscopic approach differ dramatically for some nuclei. Our calculations
indicate, in contrast to the self-consistent mean-field studies, that fission
barriers, still quite substantial for some $Z=118$ nuclei, become lower than
5.5 MeV for $Z=126$.

##  V ACKNOWLEDGMENT

M.K. and J.S. were co-financed by the National Science Centre under Contract
No. UMO-2013/08/M/ST2/00257 (LEA COPIGAL). One of the authors (P.J.) was
cofinanced by Ministry of Science and Higher Education: „Iuventus Plus” grant
Nr IP2014 016073. This research was also supported by an allocation of
advanced computing resources provided by the Swierk Computing Centre (CIS) at
the National Centre for Nuclear Research (NCBJ) (http://www.cis.gov.pl).

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Table 3:  Calculated fission barrier heights (in MeV). | N | A | $B_{f}$ | N | A | $B_{f}$ | N | A | $B_{f}$ | N | A | $B_{f}$ | N | A | $B_{f}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
|  | Z=98 |  |  | Z=99 |  |  | Z=100 |  |  | Z=101 |  |  | Z=102 |
| 134 | 232 | 2.28 |  |  |  |  |  |  |  |  |  |  |  |
| 135 | 233 | 2.74 | 135 | 234 | 2.82 |  |  |  |  |  |  |  |  |
| 136 | 234 | 2.83 | 136 | 235 | 3.30 | 136 | 236 | 2.62 |  |  |  |  |  |
| 137 | 235 | 3.45 | 137 | 236 | 4.18 | 137 | 237 | 3.30 | 137 | 238 | 3.29 |  |  |
| 138 | 236 | 3.62 | 138 | 237 | 4.37 | 138 | 238 | 3.58 | 138 | 239 | 3.33 | 138 | 240 | 2.87
| 139 | 237 | 4.64 | 139 | 238 | 5.32 | 139 | 239 | 4.64 | 139 | 240 | 4.13 | 139 | 241 | 3.81
| 140 | 238 | 4.78 | 140 | 239 | 5.23 | 140 | 240 | 4.61 | 140 | 241 | 4.05 | 140 | 242 | 3.63
| 141 | 239 | 5.86 | 141 | 240 | 6.14 | 141 | 241 | 5.60 | 141 | 242 | 4.94 | 141 | 243 | 4.47
| 142 | 240 | 5.90 | 142 | 241 | 6.01 | 142 | 242 | 5.38 | 142 | 243 | 4.85 | 142 | 244 | 4.42
| 143 | 241 | 6.71 | 143 | 242 | 7.01 | 143 | 243 | 6.23 | 143 | 244 | 5.70 | 143 | 245 | 5.18
| 144 | 242 | 6.61 | 144 | 243 | 6.72 | 144 | 244 | 6.07 | 144 | 245 | 5.59 | 144 | 246 | 5.10
| 145 | 243 | 7.35 | 145 | 244 | 7.72 | 145 | 245 | 7.09 | 145 | 246 | 6.48 | 145 | 247 | 5.90
| 146 | 244 | 6.88 | 146 | 245 | 7.25 | 146 | 246 | 6.61 | 146 | 247 | 6.23 | 146 | 248 | 5.66
| 147 | 245 | 7.41 | 147 | 246 | 7.99 | 147 | 247 | 7.32 | 147 | 248 | 6.94 | 147 | 249 | 6.39
| 148 | 246 | 6.86 | 148 | 247 | 7.50 | 148 | 248 | 6.89 | 148 | 249 | 6.72 | 148 | 250 | 6.14
| 149 | 247 | 7.08 | 149 | 248 | 7.87 | 149 | 249 | 7.40 | 149 | 250 | 7.45 | 149 | 251 | 6.92
| 150 | 248 | 6.79 | 150 | 249 | 7.53 | 150 | 250 | 6.99 | 150 | 251 | 7.09 | 150 | 252 | 6.59
| 151 | 249 | 7.36 | 151 | 250 | 8.06 | 151 | 251 | 7.60 | 151 | 252 | 7.91 | 151 | 253 | 7.42
| 152 | 250 | 6.67 | 152 | 251 | 7.42 | 152 | 252 | 6.98 | 152 | 253 | 7.38 | 152 | 254 | 6.88
| 153 | 251 | 6.26 | 153 | 252 | 6.95 | 153 | 253 | 6.55 | 153 | 254 | 7.03 | 153 | 255 | 6.53
| 154 | 252 | 5.98 | 154 | 253 | 6.66 | 154 | 254 | 6.21 | 154 | 255 | 6.69 | 154 | 256 | 6.23
| 155 | 253 | 5.62 | 155 | 254 | 5.88 | 155 | 255 | 5.71 | 155 | 256 | 6.06 | 155 | 257 | 5.81
| 156 | 254 | 5.19 | 156 | 255 | 5.69 | 156 | 256 | 5.40 | 156 | 257 | 5.82 | 156 | 258 | 5.46
| 157 | 255 | 5.00 | 157 | 256 | 5.32 | 157 | 257 | 5.14 | 157 | 258 | 5.77 | 157 | 259 | 5.59
| 158 | 256 | 4.73 | 158 | 257 | 5.04 | 158 | 258 | 4.82 | 158 | 259 | 5.36 | 158 | 260 | 5.15
| 159 | 257 | 4.99 | 159 | 258 | 5.26 | 159 | 259 | 5.08 | 159 | 260 | 5.30 | 159 | 261 | 5.30
| 160 | 258 | 4.48 | 160 | 259 | 4.63 | 160 | 260 | 4.56 | 160 | 261 | 4.95 | 160 | 262 | 5.02
| 161 | 259 | 5.06 | 161 | 260 | 5.19 | 161 | 261 | 5.17 | 161 | 262 | 5.61 | 161 | 263 | 5.46
| 162 | 260 | 4.60 | 162 | 261 | 4.71 | 162 | 262 | 4.74 | 162 | 263 | 5.23 | 162 | 264 | 4.96
| 163 | 261 | 4.41 | 163 | 262 | 4.58 | 163 | 263 | 4.54 | 163 | 264 | 4.97 | 163 | 265 | 4.71
| 164 | 262 | 4.10 | 164 | 263 | 4.20 | 164 | 264 | 4.14 | 164 | 265 | 4.56 | 164 | 266 | 4.29
| 165 | 263 | 3.97 | 165 | 264 | 3.99 | 165 | 265 | 3.85 | 165 | 266 | 4.15 | 165 | 267 | 3.86
| 166 | 264 | 3.71 | 166 | 265 | 3.78 | 166 | 266 | 3.62 | 166 | 267 | 3.92 | 166 | 268 | 3.69
| 167 | 265 | 3.71 | 167 | 266 | 3.65 | 167 | 267 | 3.51 | 167 | 268 | 3.64 | 167 | 269 | 3.52
| 168 | 266 | 3.62 | 168 | 267 | 3.50 | 168 | 268 | 3.38 | 168 | 269 | 3.55 | 168 | 270 | 3.27
| 169 | 267 | 4.38 | 169 | 268 | 3.78 | 169 | 269 | 3.84 | 169 | 270 | 3.69 | 169 | 271 | 3.51
| 170 | 268 | 3.85 | 170 | 269 | 3.52 | 170 | 270 | 3.43 | 170 | 271 | 3.35 | 170 | 272 | 3.19
| 171 | 269 | 4.81 | 171 | 270 | 4.20 | 171 | 271 | 4.36 | 171 | 272 | 3.92 | 171 | 273 | 3.93
| 172 | 270 | 4.48 | 172 | 271 | 3.79 | 172 | 272 | 3.94 | 172 | 273 | 3.50 | 172 | 274 | 3.46
| 173 | 271 | 5.13 | 173 | 272 | 4.46 | 173 | 273 | 4.62 | 173 | 274 | 4.31 | 173 | 275 | 4.08
| 174 | 272 | 5.13 | 174 | 273 | 4.31 | 174 | 274 | 4.48 | 174 | 275 | 3.99 | 174 | 276 | 3.91
| 175 | 273 | 6.00 | 175 | 274 | 5.18 | 175 | 275 | 5.37 | 175 | 276 | 4.87 | 175 | 277 | 4.72
| 176 | 274 | 5.58 | 176 | 275 | 4.89 | 176 | 276 | 5.01 | 176 | 277 | 4.60 | 176 | 278 | 4.39
| 177 | 275 | 6.63 | 177 | 276 | 6.10 | 177 | 277 | 6.01 | 177 | 278 | 5.56 | 177 | 279 | 5.41
| 178 | 276 | 6.17 | 178 | 277 | 5.41 | 178 | 278 | 5.52 | 178 | 279 | 5.04 | 178 | 280 | 4.86
| 179 | 277 | 6.72 | 179 | 278 | 6.03 | 179 | 279 | 5.99 | 179 | 280 | 5.43 | 179 | 281 | 5.37
| 180 | 278 | 6.49 | 180 | 279 | 5.73 | 180 | 280 | 5.66 | 180 | 281 | 5.10 | 180 | 282 | 5.04
| 181 | 279 | 7.85 | 181 | 280 | 6.50 | 181 | 281 | 6.41 | 181 | 282 | 5.71 | 181 | 283 | 6.22
| 182 | 280 | 6.93 | 182 | 281 | 5.99 | 182 | 282 | 5.96 | 182 | 283 | 5.13 | 182 | 284 | 5.31
| 183 | 281 | 7.65 | 183 | 282 | 6.68 | 183 | 283 | 6.65 | 183 | 284 | 5.95 | 183 | 285 | 6.11
| 184 | 282 | 7.14 | 184 | 283 | 6.19 | 184 | 284 | 6.17 | 184 | 285 | 5.36 | 184 | 286 | 5.48
| 185 | 283 | 5.73 | 185 | 284 | 4.79 | 185 | 285 | 4.70 | 185 | 286 | 4.29 | 185 | 287 | 4.29
| 186 | 284 | 5.43 | 186 | 285 | 4.54 | 186 | 286 | 4.47 | 186 | 287 | 4.02 | 186 | 288 | 3.97
| 187 | 285 | 4.59 | 187 | 286 | 3.76 | 187 | 287 | 3.60 | 187 | 288 | 3.38 | 187 | 289 | 3.20
| 188 | 286 | 4.00 | 188 | 287 | 3.30 | 188 | 288 | 3.09 | 188 | 289 | 3.00 | 188 | 290 | 2.73
| 189 | 287 | 4.13 | 189 | 288 | 3.30 | 189 | 289 | 3.10 | 189 | 290 | 2.88 | 189 | 291 | 2.48
| 190 | 288 | 3.52 | 190 | 289 | 2.79 | 190 | 290 | 2.48 | 190 | 291 | 2.42 | 190 | 292 | 2.09
| 191 | 289 | 3.53 | 191 | 290 | 2.69 | 191 | 291 | 2.57 | 191 | 292 | 2.64 | 191 | 293 | 2.24
| 192 | 290 | 3.08 | 192 | 291 | 2.17 | 192 | 292 | 2.05 | 192 | 293 | 2.13 | 192 | 294 | 1.73
Table 4:  Calculated fission barrier heights (in MeV). | N | A | $B_{f}$ | N | A | $B_{f}$ | N | A | $B_{f}$ | N | A | $B_{f}$ | N | A | $B_{f}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
|  | Z=103 |  |  | Z=104 |  |  | Z=105 |  |  | Z=106 |  |  | Z=107 |
| 139 | 242 | 3.13 |  |  |  |  |  |  |  |  |  |  |  |
| 140 | 243 | 3.14 | 140 | 244 | 2.69 |  |  |  |  |  |  |  |  |
| 141 | 244 | 4.01 | 141 | 245 | 3.58 | 141 | 246 | 2.89 |  |  |  |  |  |
| 142 | 245 | 3.92 | 142 | 246 | 3.46 | 142 | 247 | 3.09 | 142 | 248 | 2.66 |  |  |
| 143 | 246 | 4.55 | 143 | 247 | 4.16 | 143 | 248 | 3.77 | 143 | 249 | 3.37 | 143 | 250 | 3.30
| 144 | 247 | 4.66 | 144 | 248 | 4.13 | 144 | 249 | 3.75 | 144 | 250 | 3.35 | 144 | 251 | 3.31
| 145 | 248 | 5.29 | 145 | 249 | 4.95 | 145 | 250 | 4.68 | 145 | 251 | 4.21 | 145 | 252 | 4.17
| 146 | 249 | 5.19 | 146 | 250 | 4.74 | 146 | 251 | 4.56 | 146 | 252 | 4.15 | 146 | 253 | 4.16
| 147 | 250 | 6.10 | 147 | 251 | 5.47 | 147 | 252 | 5.57 | 147 | 253 | 5.01 | 147 | 254 | 5.19
| 148 | 251 | 5.79 | 148 | 252 | 5.36 | 148 | 253 | 5.62 | 148 | 254 | 4.98 | 148 | 255 | 5.08
| 149 | 252 | 6.54 | 149 | 253 | 6.16 | 149 | 254 | 6.77 | 149 | 255 | 5.98 | 149 | 256 | 6.22
| 150 | 253 | 6.28 | 150 | 254 | 5.93 | 150 | 255 | 6.43 | 150 | 256 | 5.76 | 150 | 257 | 5.92
| 151 | 254 | 7.26 | 151 | 255 | 6.93 | 151 | 256 | 7.50 | 151 | 257 | 6.85 | 151 | 258 | 7.17
| 152 | 255 | 6.81 | 152 | 256 | 6.44 | 152 | 257 | 7.04 | 152 | 258 | 6.37 | 152 | 259 | 6.70
| 153 | 256 | 6.62 | 153 | 257 | 6.36 | 153 | 258 | 7.18 | 153 | 259 | 6.58 | 153 | 260 | 6.83
| 154 | 257 | 6.45 | 154 | 258 | 6.11 | 154 | 259 | 6.99 | 154 | 260 | 6.49 | 154 | 261 | 6.70
| 155 | 258 | 6.49 | 155 | 259 | 6.14 | 155 | 260 | 7.18 | 155 | 261 | 6.61 | 155 | 262 | 6.69
| 156 | 259 | 6.30 | 156 | 260 | 5.96 | 156 | 261 | 6.82 | 156 | 262 | 6.30 | 156 | 263 | 6.53
| 157 | 260 | 6.33 | 157 | 261 | 6.01 | 157 | 262 | 6.84 | 157 | 263 | 6.37 | 157 | 264 | 6.74
| 158 | 261 | 6.10 | 158 | 262 | 5.73 | 158 | 263 | 6.56 | 158 | 264 | 6.03 | 158 | 265 | 6.44
| 159 | 262 | 6.15 | 159 | 263 | 5.68 | 159 | 264 | 6.53 | 159 | 265 | 5.97 | 159 | 266 | 6.92
| 160 | 263 | 5.89 | 160 | 264 | 5.45 | 160 | 265 | 6.31 | 160 | 266 | 5.83 | 160 | 267 | 6.72
| 161 | 264 | 6.36 | 161 | 265 | 5.91 | 161 | 266 | 6.83 | 161 | 267 | 6.45 | 161 | 268 | 7.60
| 162 | 265 | 5.83 | 162 | 266 | 5.46 | 162 | 267 | 6.40 | 162 | 268 | 5.95 | 162 | 269 | 7.20
| 163 | 266 | 5.19 | 163 | 267 | 5.14 | 163 | 268 | 5.99 | 163 | 269 | 5.69 | 163 | 270 | 6.95
| 164 | 267 | 4.83 | 164 | 268 | 4.59 | 164 | 269 | 5.37 | 164 | 270 | 5.06 | 164 | 271 | 6.27
| 165 | 268 | 4.17 | 165 | 269 | 4.13 | 165 | 270 | 4.71 | 165 | 271 | 4.50 | 165 | 272 | 5.61
| 166 | 269 | 3.97 | 166 | 270 | 3.85 | 166 | 271 | 4.43 | 166 | 272 | 4.17 | 166 | 273 | 5.30
| 167 | 270 | 3.82 | 167 | 271 | 3.67 | 167 | 272 | 4.00 | 167 | 273 | 3.85 | 167 | 274 | 4.67
| 168 | 271 | 3.41 | 168 | 272 | 3.33 | 168 | 273 | 3.70 | 168 | 274 | 3.54 | 168 | 275 | 4.38
| 169 | 272 | 3.52 | 169 | 273 | 3.44 | 169 | 274 | 3.67 | 169 | 275 | 3.44 | 169 | 276 | 3.93
| 170 | 273 | 3.19 | 170 | 274 | 3.12 | 170 | 275 | 3.37 | 170 | 276 | 3.20 | 170 | 277 | 3.72
| 171 | 274 | 3.68 | 171 | 275 | 3.65 | 171 | 276 | 3.81 | 171 | 277 | 3.56 | 171 | 278 | 4.11
| 172 | 275 | 3.30 | 172 | 276 | 3.20 | 172 | 277 | 3.36 | 172 | 278 | 3.24 | 172 | 279 | 3.70
| 173 | 276 | 4.07 | 173 | 277 | 3.83 | 173 | 278 | 4.08 | 173 | 279 | 3.89 | 173 | 280 | 4.33
| 174 | 277 | 3.67 | 174 | 278 | 3.48 | 174 | 279 | 3.72 | 174 | 280 | 3.55 | 174 | 281 | 4.10
| 175 | 278 | 4.55 | 175 | 279 | 4.55 | 175 | 280 | 4.73 | 175 | 281 | 4.71 | 175 | 282 | 5.15
| 176 | 279 | 4.00 | 176 | 280 | 4.12 | 176 | 281 | 4.10 | 176 | 282 | 4.15 | 176 | 283 | 4.64
| 177 | 280 | 4.97 | 177 | 281 | 5.24 | 177 | 282 | 5.03 | 177 | 283 | 5.29 | 177 | 284 | 5.27
| 178 | 281 | 4.46 | 178 | 282 | 4.77 | 178 | 283 | 4.51 | 178 | 284 | 4.78 | 178 | 285 | 4.65
| 179 | 282 | 5.01 | 179 | 283 | 5.19 | 179 | 284 | 5.23 | 179 | 285 | 5.26 | 179 | 286 | 5.49
| 180 | 283 | 4.68 | 180 | 284 | 4.89 | 180 | 285 | 4.75 | 180 | 286 | 5.06 | 180 | 287 | 4.99
| 181 | 284 | 5.62 | 181 | 285 | 5.69 | 181 | 286 | 5.71 | 181 | 287 | 5.83 | 181 | 288 | 5.84
| 182 | 285 | 5.00 | 182 | 286 | 5.17 | 182 | 287 | 5.13 | 182 | 288 | 5.22 | 182 | 289 | 5.27
| 183 | 286 | 5.91 | 183 | 287 | 5.89 | 183 | 288 | 5.87 | 183 | 289 | 6.01 | 183 | 290 | 6.13
| 184 | 287 | 5.31 | 184 | 288 | 5.36 | 184 | 289 | 5.34 | 184 | 290 | 5.41 | 184 | 291 | 5.52
| 185 | 288 | 4.38 | 185 | 289 | 4.27 | 185 | 290 | 4.32 | 185 | 291 | 4.26 | 185 | 292 | 4.25
| 186 | 289 | 4.17 | 186 | 290 | 4.01 | 186 | 291 | 4.11 | 186 | 292 | 4.11 | 186 | 293 | 4.26
| 187 | 290 | 2.74 | 187 | 291 | 3.05 | 187 | 292 | 3.24 | 187 | 293 | 2.99 | 187 | 294 | 3.19
| 188 | 291 | 2.76 | 188 | 292 | 2.57 | 188 | 293 | 2.83 | 188 | 294 | 2.67 | 188 | 295 | 2.91
| 189 | 292 | 2.44 | 189 | 293 | 2.23 | 189 | 294 | 2.18 | 189 | 295 | 1.88 | 189 | 296 | 1.86
| 190 | 293 | 2.20 | 190 | 294 | 1.81 | 190 | 295 | 1.92 | 190 | 296 | 1.56 | 190 | 297 | 1.61
| 191 | 294 | 2.21 | 191 | 295 | 1.89 | 191 | 296 | 1.73 | 191 | 297 | 1.48 | 191 | 298 | 1.40
| 192 | 295 | 1.70 | 192 | 296 | 1.44 | 192 | 297 | 1.37 | 192 | 298 | 1.12 | 192 | 299 | 1.06
Table 5:  Calculated fission barrier heights (in MeV). | N | A | $B_{f}$ | N | A | $B_{f}$ | N | A | $B_{f}$ | N | A | $B_{f}$ | N | A | $B_{f}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
|  | Z=108 |  |  | Z=109 |  |  | Z=110 |  |  | Z=111 |  |  | Z=112 |
| 144 | 252 | 2.72 |  |  |  |  |  |  |  |  |  |  |  |
| 145 | 253 | 3.49 | 145 | 254 | 2.72 |  |  |  |  |  |  |  |  |
| 146 | 254 | 3.58 | 146 | 255 | 3.06 | 146 | 256 | 2.47 |  |  |  |  |  |
| 147 | 255 | 4.50 | 147 | 256 | 3.71 | 147 | 257 | 3.29 | 147 | 258 | 2.60 |  |  |
| 148 | 256 | 4.48 | 148 | 257 | 3.70 | 148 | 258 | 3.32 | 148 | 259 | 2.50 | 148 | 260 | 2.21
| 149 | 257 | 5.60 | 149 | 258 | 4.88 | 149 | 259 | 4.26 | 149 | 260 | 3.33 | 149 | 261 | 2.96
| 150 | 258 | 5.29 | 150 | 259 | 4.53 | 150 | 260 | 4.05 | 150 | 261 | 3.27 | 150 | 262 | 2.77
| 151 | 259 | 6.40 | 151 | 260 | 5.53 | 151 | 261 | 5.12 | 151 | 262 | 4.10 | 151 | 263 | 3.92
| 152 | 260 | 5.98 | 152 | 261 | 5.17 | 152 | 262 | 4.68 | 152 | 263 | 3.67 | 152 | 264 | 3.47
| 153 | 261 | 6.13 | 153 | 262 | 5.00 | 153 | 263 | 4.68 | 153 | 264 | 3.73 | 153 | 265 | 3.40
| 154 | 262 | 6.07 | 154 | 263 | 4.97 | 154 | 264 | 4.50 | 154 | 265 | 3.81 | 154 | 266 | 3.34
| 155 | 263 | 6.07 | 155 | 264 | 5.01 | 155 | 265 | 4.50 | 155 | 266 | 3.65 | 155 | 267 | 3.22
| 156 | 264 | 5.93 | 156 | 265 | 4.92 | 156 | 266 | 4.39 | 156 | 267 | 3.74 | 156 | 268 | 3.26
| 157 | 265 | 6.09 | 157 | 266 | 5.29 | 157 | 267 | 4.50 | 157 | 268 | 3.65 | 157 | 269 | 3.21
| 158 | 266 | 5.83 | 158 | 267 | 5.11 | 158 | 268 | 4.38 | 158 | 269 | 3.68 | 158 | 270 | 3.31
| 159 | 267 | 6.19 | 159 | 268 | 5.85 | 159 | 269 | 5.04 | 159 | 270 | 4.52 | 159 | 271 | 3.98
| 160 | 268 | 5.93 | 160 | 269 | 5.65 | 160 | 270 | 4.86 | 160 | 271 | 4.26 | 160 | 272 | 3.78
| 161 | 269 | 6.87 | 161 | 270 | 6.72 | 161 | 271 | 5.91 | 161 | 272 | 5.58 | 161 | 273 | 5.02
| 162 | 270 | 6.46 | 162 | 271 | 6.44 | 162 | 272 | 5.62 | 162 | 273 | 5.30 | 162 | 274 | 4.72
| 163 | 271 | 6.28 | 163 | 272 | 6.48 | 163 | 273 | 5.85 | 163 | 274 | 5.70 | 163 | 275 | 5.00
| 164 | 272 | 5.52 | 164 | 273 | 5.85 | 164 | 274 | 5.22 | 164 | 275 | 5.03 | 164 | 276 | 4.46
| 165 | 273 | 5.00 | 165 | 274 | 5.56 | 165 | 275 | 4.87 | 165 | 276 | 5.05 | 165 | 277 | 4.46
| 166 | 274 | 4.62 | 166 | 275 | 5.14 | 166 | 276 | 4.47 | 166 | 277 | 4.62 | 166 | 278 | 4.01
| 167 | 275 | 4.16 | 167 | 276 | 4.90 | 167 | 277 | 4.16 | 167 | 278 | 4.61 | 167 | 279 | 3.99
| 168 | 276 | 3.80 | 168 | 277 | 4.45 | 168 | 278 | 3.73 | 168 | 279 | 4.13 | 168 | 280 | 3.78
| 169 | 277 | 3.40 | 169 | 278 | 4.14 | 169 | 279 | 3.44 | 169 | 280 | 4.35 | 169 | 281 | 3.88
| 170 | 278 | 3.20 | 170 | 279 | 3.84 | 170 | 280 | 3.29 | 170 | 281 | 4.19 | 170 | 282 | 3.74
| 171 | 279 | 3.80 | 171 | 280 | 4.09 | 171 | 281 | 3.95 | 171 | 282 | 4.74 | 171 | 283 | 4.58
| 172 | 280 | 3.31 | 172 | 281 | 3.87 | 172 | 282 | 3.72 | 172 | 283 | 4.67 | 172 | 284 | 4.34
| 173 | 281 | 4.20 | 173 | 282 | 4.55 | 173 | 283 | 4.68 | 173 | 284 | 5.32 | 173 | 285 | 5.26
| 174 | 282 | 3.88 | 174 | 283 | 4.22 | 174 | 284 | 4.40 | 174 | 285 | 5.07 | 174 | 286 | 5.03
| 175 | 283 | 5.00 | 175 | 284 | 5.37 | 175 | 285 | 5.51 | 175 | 286 | 5.83 | 175 | 287 | 6.04
| 176 | 284 | 4.50 | 176 | 285 | 4.91 | 176 | 286 | 5.04 | 176 | 287 | 5.37 | 176 | 288 | 5.60
| 177 | 285 | 5.43 | 177 | 286 | 5.74 | 177 | 287 | 5.86 | 177 | 288 | 6.39 | 177 | 289 | 6.29
| 178 | 286 | 4.93 | 178 | 287 | 5.17 | 178 | 288 | 5.33 | 178 | 289 | 5.93 | 178 | 290 | 5.87
| 179 | 287 | 5.61 | 179 | 288 | 6.03 | 179 | 289 | 5.96 | 179 | 290 | 6.60 | 179 | 291 | 6.32
| 180 | 288 | 5.29 | 180 | 289 | 5.66 | 180 | 290 | 5.61 | 180 | 291 | 6.23 | 180 | 292 | 5.94
| 181 | 289 | 6.06 | 181 | 290 | 6.19 | 181 | 291 | 6.12 | 181 | 292 | 6.59 | 181 | 293 | 6.29
| 182 | 290 | 5.47 | 182 | 291 | 5.70 | 182 | 292 | 5.63 | 182 | 293 | 6.18 | 182 | 294 | 5.89
| 183 | 291 | 6.05 | 183 | 292 | 6.46 | 183 | 293 | 6.20 | 183 | 294 | 6.75 | 183 | 295 | 6.48
| 184 | 292 | 5.61 | 184 | 293 | 5.95 | 184 | 294 | 5.68 | 184 | 295 | 6.20 | 184 | 296 | 5.91
| 185 | 293 | 4.20 | 185 | 294 | 4.51 | 185 | 295 | 4.38 | 185 | 296 | 4.79 | 185 | 297 | 4.54
| 186 | 294 | 4.23 | 186 | 295 | 4.68 | 186 | 296 | 4.50 | 186 | 297 | 5.04 | 186 | 298 | 4.74
| 187 | 295 | 3.04 | 187 | 296 | 3.09 | 187 | 297 | 3.07 | 187 | 298 | 3.73 | 187 | 299 | 3.39
| 188 | 296 | 2.83 | 188 | 297 | 3.11 | 188 | 298 | 3.01 | 188 | 299 | 3.76 | 188 | 300 | 3.44
| 189 | 297 | 1.75 | 189 | 298 | 1.80 | 189 | 299 | 1.79 | 189 | 300 | 2.20 | 189 | 301 | 1.92
| 190 | 298 | 1.44 | 190 | 299 | 1.74 | 190 | 300 | 1.58 | 190 | 301 | 2.28 | 190 | 302 | 2.02
| 191 | 299 | 1.30 | 191 | 300 | 1.21 | 191 | 301 | 1.28 | 191 | 302 | 1.05 | 191 | 303 | 1.07
| 192 | 300 | 0.75 | 192 | 301 | 0.74 | 192 | 302 | 0.76 | 192 | 303 | 0.99 | 192 | 304 | 0.76
Table 6:  Calculated fission barrier heights (in MeV). | N | A | $B_{f}$ | N | A | $B_{f}$ | N | A | $B_{f}$ | N | A | $B_{f}$ | N | A | $B_{f}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
|  | Z=113 |  |  | Z=114 |  |  | Z=115 |  |  | Z=116 |  |  | Z=117 |
| 149 | 262 | 2.02 |  |  |  |  |  |  |  |  |  |  |  |
| 150 | 263 | 2.02 | 150 | 264 | 1.72 |  |  |  |  |  |  |  |  |
| 151 | 264 | 2.95 | 151 | 265 | 2.75 | 151 | 266 | 1.54 |  |  |  |  |  |
| 152 | 265 | 2.44 | 152 | 266 | 2.27 | 152 | 267 | 1.17 | 152 | 268 | 1.08 |  |  |
| 153 | 266 | 2.33 | 153 | 267 | 2.15 | 153 | 268 | 0.90 | 153 | 269 | 0.96 | 153 | 270 | 0.76
| 154 | 267 | 2.25 | 154 | 268 | 2.14 | 154 | 269 | 1.34 | 154 | 270 | 1.01 | 154 | 271 | 0.87
| 155 | 268 | 2.44 | 155 | 269 | 2.13 | 155 | 270 | 2.14 | 155 | 271 | 1.39 | 155 | 272 | 1.93
| 156 | 269 | 2.34 | 156 | 270 | 2.27 | 156 | 271 | 2.29 | 156 | 272 | 1.72 | 156 | 273 | 1.99
| 157 | 270 | 2.55 | 157 | 271 | 2.36 | 157 | 272 | 2.81 | 157 | 273 | 2.24 | 157 | 274 | 2.58
| 158 | 271 | 2.66 | 158 | 272 | 2.50 | 158 | 273 | 2.89 | 158 | 274 | 2.37 | 158 | 275 | 2.91
| 159 | 272 | 3.65 | 159 | 273 | 3.25 | 159 | 274 | 3.67 | 159 | 275 | 3.13 | 159 | 276 | 3.33
| 160 | 273 | 3.45 | 160 | 274 | 3.39 | 160 | 275 | 3.81 | 160 | 276 | 3.27 | 160 | 277 | 3.69
| 161 | 274 | 4.58 | 161 | 275 | 4.53 | 161 | 276 | 4.88 | 161 | 277 | 4.45 | 161 | 278 | 4.74
| 162 | 275 | 4.41 | 162 | 276 | 4.35 | 162 | 277 | 4.68 | 162 | 278 | 4.18 | 162 | 279 | 4.35
| 163 | 276 | 4.88 | 163 | 277 | 4.75 | 163 | 278 | 5.30 | 163 | 279 | 4.77 | 163 | 280 | 5.10
| 164 | 277 | 4.44 | 164 | 278 | 4.32 | 164 | 279 | 4.79 | 164 | 280 | 4.34 | 164 | 281 | 4.66
| 165 | 278 | 4.58 | 165 | 279 | 4.29 | 165 | 280 | 4.87 | 165 | 281 | 4.36 | 165 | 282 | 4.47
| 166 | 279 | 4.23 | 166 | 280 | 3.99 | 166 | 281 | 4.56 | 166 | 282 | 4.16 | 166 | 283 | 4.53
| 167 | 280 | 4.67 | 167 | 281 | 4.37 | 167 | 282 | 4.96 | 167 | 283 | 4.52 | 167 | 284 | 4.77
| 168 | 281 | 4.34 | 168 | 282 | 4.05 | 168 | 283 | 4.81 | 168 | 284 | 4.48 | 168 | 285 | 4.69
| 169 | 282 | 4.89 | 169 | 283 | 4.52 | 169 | 284 | 5.29 | 169 | 285 | 4.83 | 169 | 286 | 5.23
| 170 | 283 | 4.46 | 170 | 284 | 4.48 | 170 | 285 | 4.98 | 170 | 286 | 4.70 | 170 | 287 | 5.30
| 171 | 284 | 5.19 | 171 | 285 | 5.16 | 171 | 286 | 5.70 | 171 | 287 | 5.76 | 171 | 288 | 6.21
| 172 | 285 | 4.98 | 172 | 286 | 4.83 | 172 | 287 | 5.56 | 172 | 288 | 5.45 | 172 | 289 | 5.95
| 173 | 286 | 5.74 | 173 | 287 | 5.76 | 173 | 288 | 6.21 | 173 | 289 | 6.18 | 173 | 290 | 6.81
| 174 | 287 | 5.54 | 174 | 288 | 5.52 | 174 | 289 | 6.02 | 174 | 290 | 5.93 | 174 | 291 | 6.46
| 175 | 288 | 6.43 | 175 | 289 | 6.43 | 175 | 290 | 6.68 | 175 | 291 | 6.67 | 175 | 292 | 7.04
| 176 | 289 | 6.28 | 176 | 290 | 6.04 | 176 | 291 | 6.55 | 176 | 292 | 6.31 | 176 | 293 | 6.56
| 177 | 290 | 6.95 | 177 | 291 | 6.67 | 177 | 292 | 7.46 | 177 | 293 | 6.82 | 177 | 294 | 7.31
| 178 | 291 | 6.61 | 178 | 292 | 6.58 | 178 | 293 | 6.80 | 178 | 294 | 6.37 | 178 | 295 | 6.64
| 179 | 292 | 7.26 | 179 | 293 | 6.75 | 179 | 294 | 7.08 | 179 | 295 | 6.64 | 179 | 296 | 6.88
| 180 | 293 | 6.82 | 180 | 294 | 6.45 | 180 | 295 | 6.69 | 180 | 296 | 6.25 | 180 | 297 | 6.32
| 181 | 294 | 6.93 | 181 | 295 | 6.64 | 181 | 296 | 7.00 | 181 | 297 | 6.51 | 181 | 298 | 6.70
| 182 | 295 | 6.71 | 182 | 296 | 6.29 | 182 | 297 | 6.48 | 182 | 298 | 6.10 | 182 | 299 | 6.12
| 183 | 296 | 7.13 | 183 | 297 | 6.58 | 183 | 298 | 6.75 | 183 | 299 | 6.35 | 183 | 300 | 6.32
| 184 | 297 | 6.63 | 184 | 298 | 6.10 | 184 | 299 | 6.23 | 184 | 300 | 5.84 | 184 | 301 | 5.81
| 185 | 298 | 5.36 | 185 | 299 | 4.72 | 185 | 300 | 4.97 | 185 | 301 | 4.39 | 185 | 302 | 4.27
| 186 | 299 | 5.43 | 186 | 300 | 4.92 | 186 | 301 | 5.01 | 186 | 302 | 4.64 | 186 | 303 | 4.61
| 187 | 300 | 4.11 | 187 | 301 | 3.52 | 187 | 302 | 3.85 | 187 | 303 | 3.27 | 187 | 304 | 3.32
| 188 | 301 | 4.14 | 188 | 302 | 3.66 | 188 | 303 | 3.74 | 188 | 304 | 3.40 | 188 | 305 | 3.34
| 189 | 302 | 2.67 | 189 | 303 | 2.26 | 189 | 304 | 2.53 | 189 | 305 | 2.04 | 189 | 306 | 2.28
| 190 | 303 | 2.70 | 190 | 304 | 2.36 | 190 | 305 | 2.34 | 190 | 306 | 2.16 | 190 | 307 | 2.07
| 191 | 304 | 1.30 | 191 | 305 | 0.78 | 191 | 306 | 1.53 | 191 | 307 | 0.90 | 191 | 308 | 1.25
| 192 | 305 | 1.28 | 192 | 306 | 0.91 | 192 | 307 | 1.23 | 192 | 308 | 0.75 | 192 | 309 | 1.18
Table 7:  Calculated fission barrier heights (in MeV). | N | A | $B_{f}$ | N | A | $B_{f}$ | N | A | $B_{f}$ | N | A | $B_{f}$ | N | A | $B_{f}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
|  | Z=118 |  |  | Z=119 |  |  | Z=120 |  |  | Z=121 |  |  | Z=122 |
| 154 | 272 | 0.66 |  |  |  |  |  |  |  |  |  |  |  |
| 155 | 273 | 1.39 | 155 | 274 | 1.82 |  |  |  |  |  |  |  |  |
| 156 | 274 | 1.41 | 156 | 275 | 1.83 | 156 | 276 | 1.38 |  |  |  |  |  |
| 157 | 275 | 2.30 | 157 | 276 | 2.26 | 157 | 277 | 1.79 | 157 | 278 | 1.96 |  |  |
| 158 | 276 | 2.25 | 158 | 277 | 2.25 | 158 | 278 | 1.88 | 158 | 279 | 2.11 | 158 | 280 | 1.47
| 159 | 277 | 3.06 | 159 | 278 | 2.78 | 159 | 279 | 2.39 | 159 | 280 | 2.72 | 159 | 281 | 1.94
| 160 | 278 | 3.15 | 160 | 279 | 2.79 | 160 | 280 | 2.44 | 160 | 281 | 2.75 | 160 | 282 | 2.15
| 161 | 279 | 4.30 | 161 | 280 | 3.49 | 161 | 281 | 3.23 | 161 | 282 | 3.66 | 161 | 283 | 3.07
| 162 | 280 | 3.94 | 162 | 281 | 3.27 | 162 | 282 | 3.07 | 162 | 283 | 3.50 | 162 | 284 | 2.94
| 163 | 281 | 4.32 | 163 | 282 | 3.96 | 163 | 283 | 3.89 | 163 | 284 | 4.56 | 163 | 285 | 3.88
| 164 | 282 | 4.13 | 164 | 283 | 3.68 | 164 | 284 | 3.57 | 164 | 285 | 4.16 | 164 | 286 | 3.61
| 165 | 283 | 3.79 | 165 | 284 | 3.87 | 165 | 285 | 3.65 | 165 | 286 | 4.67 | 165 | 287 | 4.46
| 166 | 284 | 3.64 | 166 | 285 | 3.67 | 166 | 286 | 3.43 | 166 | 287 | 4.36 | 166 | 288 | 3.91
| 167 | 285 | 4.09 | 167 | 286 | 4.43 | 167 | 287 | 4.08 | 167 | 288 | 4.40 | 167 | 289 | 4.06
| 168 | 286 | 4.01 | 168 | 287 | 4.45 | 168 | 288 | 4.21 | 168 | 289 | 4.37 | 168 | 290 | 4.01
| 169 | 287 | 5.03 | 169 | 288 | 5.21 | 169 | 289 | 5.01 | 169 | 290 | 5.35 | 169 | 291 | 4.83
| 170 | 288 | 5.05 | 170 | 289 | 5.23 | 170 | 290 | 5.01 | 170 | 291 | 5.41 | 170 | 292 | 4.80
| 171 | 289 | 6.03 | 171 | 290 | 6.26 | 171 | 291 | 6.08 | 171 | 292 | 6.56 | 171 | 293 | 5.96
| 172 | 290 | 5.75 | 172 | 291 | 5.75 | 172 | 292 | 5.48 | 172 | 293 | 5.90 | 172 | 294 | 5.36
| 173 | 291 | 6.40 | 173 | 292 | 6.55 | 173 | 293 | 6.06 | 173 | 294 | 6.15 | 173 | 295 | 5.54
| 174 | 292 | 6.09 | 174 | 293 | 6.21 | 174 | 294 | 5.62 | 174 | 295 | 5.89 | 174 | 296 | 5.28
| 175 | 293 | 6.62 | 175 | 294 | 6.95 | 175 | 295 | 6.28 | 175 | 296 | 6.00 | 175 | 297 | 5.48
| 176 | 294 | 6.09 | 176 | 295 | 6.32 | 176 | 296 | 5.79 | 176 | 297 | 5.69 | 176 | 298 | 5.10
| 177 | 295 | 6.64 | 177 | 296 | 6.71 | 177 | 297 | 6.02 | 177 | 298 | 5.90 | 177 | 299 | 5.36
| 178 | 296 | 6.12 | 178 | 297 | 6.20 | 178 | 298 | 5.56 | 178 | 299 | 5.38 | 178 | 300 | 4.86
| 179 | 297 | 6.21 | 179 | 298 | 6.20 | 179 | 299 | 5.57 | 179 | 300 | 5.38 | 179 | 301 | 4.75
| 180 | 298 | 5.79 | 180 | 299 | 5.72 | 180 | 300 | 5.08 | 180 | 301 | 4.82 | 180 | 302 | 4.26
| 181 | 299 | 6.02 | 181 | 300 | 5.88 | 181 | 301 | 5.24 | 181 | 302 | 4.99 | 181 | 303 | 4.43
| 182 | 300 | 5.52 | 182 | 301 | 5.38 | 182 | 302 | 4.71 | 182 | 303 | 4.37 | 182 | 304 | 3.78
| 183 | 301 | 5.71 | 183 | 302 | 5.50 | 183 | 303 | 4.81 | 183 | 304 | 4.27 | 183 | 305 | 3.70
| 184 | 302 | 5.20 | 184 | 303 | 4.98 | 184 | 304 | 4.30 | 184 | 305 | 3.64 | 184 | 306 | 3.16
| 185 | 303 | 3.93 | 185 | 304 | 3.82 | 185 | 305 | 3.02 | 185 | 306 | 2.96 | 185 | 307 | 2.27
| 186 | 304 | 4.03 | 186 | 305 | 3.85 | 186 | 306 | 3.14 | 186 | 307 | 2.60 | 186 | 308 | 2.08
| 187 | 305 | 2.80 | 187 | 306 | 2.78 | 187 | 307 | 2.17 | 187 | 308 | 2.15 | 187 | 309 | 1.66
| 188 | 306 | 2.78 | 188 | 307 | 2.63 | 188 | 308 | 1.95 | 188 | 309 | 1.84 | 188 | 310 | 1.50
| 189 | 307 | 1.78 | 189 | 308 | 2.03 | 189 | 309 | 1.51 | 189 | 310 | 1.50 | 189 | 311 | 1.31
| 190 | 308 | 1.57 | 190 | 309 | 1.97 | 190 | 310 | 1.39 | 190 | 311 | 1.45 | 190 | 312 | 1.15
| 191 | 309 | 0.79 | 191 | 310 | 1.67 | 191 | 311 | 1.05 | 191 | 312 | 1.20 | 191 | 313 | 0.68
| 192 | 310 | 0.80 | 192 | 311 | 1.63 | 192 | 312 | 1.05 | 192 | 313 | 1.28 | 192 | 314 | 0.80
Table 8:  Calculated fission barrier heights (in MeV). | N | A | $B_{f}$ | N | A | $B_{f}$ | N | A | $B_{f}$ | N | A | $B_{f}$
---|---|---|---|---|---|---|---|---|---|---|---|---
|  | Z=123 |  |  | Z=124 |  |  | Z=125 |  |  | Z=126 |
| 159 | 282 | 2.14 |  |  |  |  |  |  |  |  |
| 160 | 283 | 2.25 | 160 | 284 | 1.78 |  |  |  |  |  |
| 161 | 284 | 3.01 | 161 | 285 | 2.51 | 161 | 286 | 2.47 |  |  |
| 162 | 285 | 2.87 | 162 | 286 | 2.44 | 162 | 287 | 2.44 | 162 | 288 | 1.89
| 163 | 286 | 4.03 | 163 | 287 | 3.46 | 163 | 288 | 3.18 | 163 | 289 | 2.64
| 164 | 287 | 3.69 | 164 | 288 | 3.24 | 164 | 289 | 3.18 | 164 | 290 | 2.58
| 165 | 288 | 4.75 | 165 | 289 | 4.43 | 165 | 290 | 4.43 | 165 | 291 | 3.85
| 166 | 289 | 4.15 | 166 | 290 | 3.78 | 166 | 291 | 3.76 | 166 | 292 | 3.24
| 167 | 290 | 4.43 | 167 | 291 | 4.08 | 167 | 292 | 4.14 | 167 | 293 | 3.69
| 168 | 291 | 4.08 | 168 | 292 | 3.54 | 168 | 293 | 3.64 | 168 | 294 | 3.15
| 169 | 292 | 5.10 | 169 | 293 | 4.44 | 169 | 294 | 4.34 | 169 | 295 | 3.48
| 170 | 293 | 4.94 | 170 | 294 | 4.38 | 170 | 295 | 4.35 | 170 | 296 | 3.49
| 171 | 294 | 5.96 | 171 | 295 | 5.44 | 171 | 296 | 5.47 | 171 | 297 | 4.62
| 172 | 295 | 5.52 | 172 | 296 | 5.01 | 172 | 297 | 5.06 | 172 | 298 | 4.20
| 173 | 296 | 5.52 | 173 | 297 | 5.10 | 173 | 298 | 5.30 | 173 | 299 | 4.84
| 174 | 297 | 5.23 | 174 | 298 | 4.81 | 174 | 299 | 5.02 | 174 | 300 | 4.52
| 175 | 298 | 5.16 | 175 | 299 | 4.82 | 175 | 300 | 5.13 | 175 | 301 | 4.65
| 176 | 299 | 4.82 | 176 | 300 | 4.51 | 176 | 301 | 4.76 | 176 | 302 | 4.37
| 177 | 300 | 5.26 | 177 | 301 | 4.92 | 177 | 302 | 5.14 | 177 | 303 | 4.75
| 178 | 301 | 4.75 | 178 | 302 | 4.48 | 178 | 303 | 4.66 | 178 | 304 | 4.28
| 179 | 302 | 4.83 | 179 | 303 | 4.55 | 179 | 304 | 4.68 | 179 | 305 | 4.33
| 180 | 303 | 4.07 | 180 | 304 | 3.84 | 180 | 305 | 4.03 | 180 | 306 | 3.66
| 181 | 304 | 3.90 | 181 | 305 | 3.39 | 181 | 306 | 3.45 | 181 | 307 | 2.96
| 182 | 305 | 3.39 | 182 | 306 | 2.95 | 182 | 307 | 3.17 | 182 | 308 | 2.79
| 183 | 306 | 3.54 | 183 | 307 | 2.78 | 183 | 308 | 3.02 | 183 | 309 | 2.21
| 184 | 307 | 2.91 | 184 | 308 | 2.25 | 184 | 309 | 2.53 | 184 | 310 | 2.08
| 185 | 308 | 2.20 | 185 | 309 | 2.07 | 185 | 310 | 2.40 | 185 | 311 | 2.05
| 186 | 309 | 2.01 | 186 | 310 | 1.96 | 186 | 311 | 2.37 | 186 | 312 | 1.90
| 187 | 310 | 1.89 | 187 | 311 | 1.69 | 187 | 312 | 1.90 | 187 | 313 | 1.39
| 188 | 311 | 1.85 | 188 | 312 | 1.62 | 188 | 313 | 1.76 | 188 | 314 | 1.26
| 189 | 312 | 1.47 | 189 | 313 | 1.16 | 189 | 314 | 1.36 | 189 | 315 | 0.90
| 190 | 313 | 1.31 | 190 | 314 | 0.98 | 190 | 315 | 1.10 | 190 | 316 | 0.80
| 191 | 314 | 0.90 | 191 | 315 | 0.68 | 191 | 316 | 0.91 | 191 | 317 | 0.81
| 192 | 315 | 0.76 | 192 | 316 | 0.62 | 192 | 317 | 0.84 | 192 | 318 | 0.70