https://ja.overleaf.com/project/61bac26ebaa5e200a37e7ab8

# Effects of triaxiality and pairing interaction on fission barriers of
actinide nuclei

Taiki Kouno  Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro, Tokyo
152-8550, Japan  Chikako Ishizuka111Corresponding author Tokyo Institute of
Technology, 2-12-1 Ookayama, Meguro, Tokyo 152-8550, Japan
<EMAIL_ADDRESS>Kazuki Fujio  Tokyo Institute of Technology,
2-12-1 Ookayama, Meguro, Tokyo 152-8550, Japan  Tsunenori Inakura  Tokyo
Institute of Technology, 2-12-1 Ookayama, Meguro, Tokyo 152-8550, Japan
Satoshi Chiba  Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro, Tokyo
152-8550, Japan

(Day Month Year; Day Month Year)

###### Abstract

We employ the density-dependent relativistic mean-field theory to study how
the triaxiality and pairing interaction affect the inner fission barriers of
actinide nuclei. It was found that the triaxiality reduced the inner barriers
and improved agreement with experimental values for many actinides. However,
about 1-2 MeV discrepancy to the experimental values still remained for some
of the considered nuclei. Such a discrepancy could be made further smaller by
increasing the BCS pairing strength parameter. In this work, we demonstrated
that adjusting the paring strength was effective in reproducing the
experimental inner fission barriers as well as “pairing rotational energy” and
binding energy in a consistent manner for nuclei where the effect of the
triaxiality on the inner barriers was significant.

###### keywords:

nuclear fission; pairing rotation; inner fission barrier

25.85.-w, 24.10.Jv, 21.10.Gv,

##  1 Introduction

Nuclear fission is a phenomenon in which a nucleus transforms into two fission
fragments while emitting occasionary a few neutrons or light-charged
particles. Understanding this phenomenon is essential for both applications
such as nuclear technology and fundamental science such as nucleosynthesis in
the cosmos.

Bohr and Wheeler’s pioneering work on nuclear fission was done based on the
liquid-drop model, and there existed only one fission barrier or saddle point,
corresponding to the activation energy of the chemical reaction, in the
potential energy characterized by deformation coordinates of nuclei [1].
Recent abundant experiments revealed that the potential energy of a fissioning
system had typically two fission barriers which were classified as “inner” and
“outer” fission barrier [2]. Nuclear fission theory had been advanced steadly
by introduction of deformed single-particle orbits and a shell correction
proposed by Strutinsky [3, 4], which could reproduce such a double-hamped
structure of the potential energy surface. Furthermore, microscopic theories,
both non-relativistic and relativistic ones, are now actively applied to the
study of nuclear fission.

Among the fission barriers, the inner one is a crucial physical quantity used
to determine the neutron fission cross-section of actinide nuclei since it is
higher than the outer one for most of the actinides important for energy
application[5]. Traditionally, the inner fission barriers have been calculated
by assuming axial symmetry in many works because of the low computational
cost; typical codes used in applications, such as SkyAx [6] and two-center
shell model [7], assume the axial symmetry. However, the inner barriers of
actinides calculated by assuming the axial symmetry ended up as an
overestimation by about $2\sim 5$ MeV compared to experimental values [8, 9].
On the other hand, the role of the triaxiality for the inner barrier has been
theoretically examined in a few decades. Many of such works used the
macroscopic-microscopic method [10, 11] and non-relativistic energy density
functionals based on Skyrme force and Gogny force [12, 13]. Nowadays a Chinese
group also found that the triaxiality suppresses the inner barrier height by
about 2 MeV using the relativistic mean-field (RMF) + point coupling model
[14, 15]. Although the influence of the triaxiality on the inner fission
barriers varies from theory to theory, we cannot ignore definitely the
influence brought by the triaxial deformation.

Our previous study [16] found that increasing the pairing strength could
enhance the agreement of the calculated inner fission barriers with
experimental data on the assumption of the axial symmetry. We evaluated the
pairing strength thus obtained by using the concept of pairing rotation, which
was an excellent observable to determined the pairing correlation because it
could avoid calculations for odd nuclei and maintain time-reversal symmetry
[17, 18]. Then, it is the purpose of the present study to evaluate the inner
fission barriers of actinide nuclei by taking account of the triaxiality using
density-dependence relativistic mean-field theory to see how the interplay
between the triaxiality and pairing strength can explain the systematical
trends of the inner barrier of actinide nuclei.

Our paper is organized as follows. Section 2 introduces the density-dependence
relativistic mean-field theory and pairing rotation. In subsection 3.1, we
first examine the changes in the inner fission barrier obtained with the
assumptions of axial symmetry and triaxiality for the case of
${}^{240}{\rm{Pu}}$ as an example. In subsection 3.2, we show the results of a
broader region of actinide nuclei. We also discuss the effects of pairing
interactions under the triaxiality on the inner barrier. Especially we address
the necessity of increasing pairing strength to improve reproducibility of the
calculated barriers with experimental values. Then we summarize results of our
study in section 4.

##  2 Method

###  2.1 Density-dependent relativistic mean-field theory

We used two density-dependent relativistic mean-field theories in this study:
density-dependent meson-exchange model (DD-ME2) and density-dependent point-
coupling interaction (DD-PC1) [19, 20, 21, 22].

Firstly, the Lagrangian in DD-ME2 is as follows [20];

$\displaystyle\mathcal{L}$ $\displaystyle=$
$\displaystyle\bar{\psi}(i\gamma\cdot\partial-m)\psi+\frac{1}{2}(\partial\sigma)^{2}-\frac{1}{2}m_{\sigma}^{2}\sigma^{2}-\frac{1}{4}\Omega_{\mu\nu}\Omega^{\mu\nu}+\frac{1}{2}m_{\omega}^{2}\omega_{\mu}\omega^{\mu}-\frac{1}{4}\vec{R}_{\mu\nu}\cdot\vec{R}^{\mu\nu}$
(1) $\displaystyle+$
$\displaystyle\frac{1}{2}m_{\rho}^{2}\vec{\rho}_{\mu}\cdot\vec{\rho}^{\mu}-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-g_{\sigma}(\rho)\sigma\bar{\psi}\psi-
g_{\omega}(\rho)\bar{\psi}\gamma^{\mu}\psi\omega_{\mu}-g_{\rho}(\rho)\bar{\psi}\vec{\tau}\gamma^{\mu}\psi\cdot\vec{\rho}$
$\displaystyle-$ $\displaystyle e\bar{\psi}\gamma^{\mu}\psi A_{\mu}$
$\displaystyle\Omega_{\mu\nu}$ $\displaystyle=$
$\displaystyle\partial_{\mu}\omega_{\nu}-\partial_{\nu}\omega_{\mu},~{}\vec{R}_{\mu\nu}=\partial_{\mu}\vec{\rho}_{\nu}-\partial_{\mu}\vec{\rho}_{\mu},~{}F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$
(2) $\displaystyle g_{i}(\rho)$ $\displaystyle=$ $\displaystyle
g_{i}(\rho_{sat})f_{i}(x),~{}\mbox{for}~{}i=\sigma,\omega,~{}f_{i}(x)=a_{i}\frac{1+b_{i}(x+d_{i})^{2}}{1+c_{i}(x+d_{i})^{2}}$
(3) $\displaystyle g_{\rho}(\rho)$ $\displaystyle=$ $\displaystyle
g_{\rho}(\rho_{sat})\exp\left[-a_{\rho}(x-1)\right],~{}x=\rho/\rho_{sat},~{}\rho_{sat}=0.152{\rm{fm}^{-3}}\
.$ (4)

Here, $\psi$ denotes the four-component nucleon Dirac field,
$\sigma,~{}\omega^{\mu},~{}{\bm{\rho}}^{\mu}$ and $A^{\mu}$ indicate scalar-
isoscalar, vector-isoscalar, vector-isovector and the photon field,
respectively. The symbol $m$ designates the nucleon mass, while
$m_{\sigma},~{}m_{\omega},~{}m_{\rho}$ denote the meson masses, and
$g_{\sigma}(\rho),~{}g_{\omega}(\rho),~{}g_{\rho}(\rho)$ represent the
density-dependent meson-nucleon coupling functions. The symbol $\rho_{sat}$ is
the saturation density of nuclear matter. In the relativistic mean-field
approach, we ignore the fluctuation of the meson field and the negative energy
components (so-called no-sea approximation). Furthermore, one of the reasons
why the coupling function has density-dependence is to reproduce the nuclear
matter properties calculated by Relativistic Brueckner-Hartree-Fock [23].

Parameter values used in DD-ME2 [19, 20] $m_{i}({\rm{MeV}})$
$g_{i}(\rho_{sat})$ $a_{i}$ $b_{i}$ $c_{i}$ $d_{i}$ $i=\sigma$ $550.1238$
$10.5396$ $1.3881$ $1.0943$ $1.7057$ $0.4421$ $i=\omega$ $783.0000$ $13.0189$
$1.3892$ $0.9240$ $1.4620$ $0.4775$ $i=\rho$ $763.0000$ $\hphantom{0}3.6836$
$0.5647$ - - -

Table 1 shows parameters in DD-ME2, which were determined to reproduce the
binding energies, charge radii, and neutron radii of spherical nuclei [19,
20]. This parameter set yields the followng nuclear-matter properties; the
binding energy per nucleon is $16.24~{}{\rm{MeV}}$, the incompressibility is
$250~{}{\rm{MeV}}$, and the symmetry energy at saturation density is
$32.3~{}{\rm{MeV}}$.

Nextly, the Lagrangian in DD-PC1 is given as follows [21, 22];

$\displaystyle\mathcal{L}$ $\displaystyle=$
$\displaystyle\bar{\psi}(i\gamma\cdot\partial-m)\psi-\frac{1}{2}\alpha_{S}(\rho)(\bar{\psi}\psi)(\bar{\psi}\psi)-\frac{1}{2}\alpha_{V}(\rho)(\bar{\psi}\gamma^{\mu}\psi)(\bar{\psi}\gamma_{\mu}\psi)$
(5) $\displaystyle-$
$\displaystyle\frac{1}{2}\alpha_{TV}(\rho)(\bar{\psi}\vec{\tau}\gamma^{\mu}\psi)(\bar{\psi}\vec{\tau}\gamma_{\mu}\psi)-\frac{1}{2}\delta_{S}(\partial_{\nu}\bar{\psi}\psi)(\partial^{\nu}\bar{\psi}\psi)-e\bar{\psi}\gamma\cdot
A\frac{1-\tau_{3}}{2}\psi\ ,$ $\displaystyle\alpha_{i}(\rho)$ $\displaystyle=$
$\displaystyle
a_{i}+(b_{i}+c_{i}x)e^{-d_{i}x},~{}(i=S,V,TV),~{}x=\rho/\rho_{sat},~{}\rho_{sat}=0.152\>{\rm{fm}^{-3}}\
.$ (6)

Here, $m$ is the nucleon mass, while
$\alpha_{S}(\rho),~{}\alpha_{V}(\rho),~{}\alpha_{TV}(\rho)$ represent the
density-dependent point coupling functions. The symbol $\psi$ represents the
nucleon four-component Dirac field. The DD-PC1 treats the nucleon Dirac field
in the same manner as DD-ME2.

Parameter values used in DD-PC1 [19, 21, 22].  $a_{i}({\rm{fm}}^{2})$
$b_{i}({\rm{fm}}^{2})$ $c_{i}({\rm{fm}}^{2})$ $d_{i}$ $i=S$
$-10.0462\hphantom{0}$ $-9.1504$ $-6.4273$ $1.3724$ $i=V$
$\hphantom{000}5.95195$ $\hphantom{00}8.8637$ $\hphantom{00}0.0\hphantom{000}$
$0.6584$ 0$i=TV$ $\hphantom{000}0.0\hphantom{0000}$ $\hphantom{00}1.8360$
$\hphantom{00}0.0\hphantom{000}$ $0.6403$

The parameters of DD-PC1 shown in Table 2, were determined to reproduce
properties of infinite and semi-infinite nuclear matter, and the binding
energies of axially symmetric deformed nuclei in the mass regions $A\approx
150-180$ and $A\approx 230-250$ [19, 21, 22]. Here, the following nuclear-
matter properties were obtained; the binding energy per nucleon is 16.14 MeV,
the incompressibility is 250.89 MeV, and the symmetry energy at saturation
density is 32.3 MeV.

In these models, the total energy is defined as $E=E_{RMF}+E_{pair}+E_{cm}$.
The $E_{RMF}$ can be calculated by volume-integration of the hamiltonian
density obtained from the Lagrangian density [24]. The center-of-mass energy
$E_{cm}$ is described as $E_{cm}=-{{\bm{P}}_{cm}^{2}}/({2Am})$, where
${\bm{P}}_{cm}$ is the center-of-mass momentum. The pairing energy $E_{pair}$
is obtained as
$E_{pair}=-(G/4)\mathrm{Tr}\left(\kappa^{\dagger}\kappa\right)$, where $G$
denotes the pairing strength and $\kappa$ the pair tensor deternined by the
relativistic Hartree-Bogoliubov calculation[19, 25]. The pairing strength was
determined as $G=728~{}{\rm{MeV\cdot fm^{3}}}$ to reproduce the pairing gap in
symmetric nuclear matter[19]. While there are several types of the pairing
correlation, we use the simplest monopole pairing at the present calculations
because complicated form of the pairing correlation introduces additional
parameters and induces the further model dependence. We used the deformed
harmonic-oscillator basis in the density-dependent models. As for the details
of the numerical method, please refer to a reference[19].

We imposed constraints on the axial and triaxial mass quadrupole moments to
investigate the total energy dependence on the deformation covering the inner
fission barrier. We defined the Hamiltonian as an unrestricted variation of
the following function:

$\displaystyle\braket{H}+\sum_{\mu=0,2}C_{2\mu}\left(\braket{\hat{Q}_{2\mu}}-q_{2\mu}\right)^{2}\
,$ (7)

where $\braket{H}$ is total energy, while $\braket{\hat{Q}_{2\mu}}$ denotes
the expectation values of the mass quadrupole operators defined as follows:

$\displaystyle\hat{Q}_{20}=2z^{2}-x^{2}-y^{2},~{}\hat{Q}_{22}=x^{2}-y^{2}.$
(8)

Furthermore, $q_{2\mu}$ define the constrained values of multipole moment, and
$C_{2\mu}$ are the corresponding stiffness constants [19, 26].

###  2.2 Pairing rotation

In this work, we will improve the reproducibility of the inner fission
barriers by increasing the strength of the pairing strength under the triaxial
deformation. Generally, the pairing correlation had been evaluated in the past
by using, as mentioned above, the pairing gap obtained from odd-even
staggering of binding energy or neutron separation energy. Calculation of the
pairing gap, however, suffers from calculations for odd nuclei since it
involes procedure such as blocking method, which has an ambiguity originating
from the mean-field part of the theory.

On the other hand, the pairing rotational energy can maintain the time-
reversal symmetry and avoid the calculation of odd nuclei because it can be
determined only by the binding energies of even-even nuclei. Therefore, in
recent years, pairing rotation has attracted attention as an effective
observable to determine pairing interaction. Pair correlation breaks the U(1)
gauge symmetry, and it derives the pairing rotational energy defined in the
3rd term of r.h.s. of the following equation; [17, 18]:

$\displaystyle E(N,Z_{0})=E(N_{0},Z_{0})+\lambda_{n}(N_{0},Z_{0})\Delta
N+\frac{(\Delta N)^{2}}{2\mathcal{I}_{nn}(N_{0},Z_{0})}\ ,$ (9)

where $E(N_{0},Z_{0})$ denotes ground-state energy for neutron number of
$N_{0}$ and proton number of $Z_{0}$ nucleus, $\Delta N=N-N_{0},\
\lambda_{n}(N_{0},Z_{0})=dE/dN|_{N=N_{0},Z=Z_{0}}$ is the chemical potential,
and the second-order term is the pairing-rotational energy with the pairing
moment-of-inertia
$\mathcal{I}_{nn}(N_{0},Z_{0})^{-1}=d^{2}E/dN^{2}|_{N=N_{0},Z=Z_{0}}$. The
experimental pairing rotational energy was obtained using the binding energies
of even-even nuclei taken from AME2016 [27].

###  2.3 Non-relativistic mean-field model

We also performed the same calculations with non-relativistic mean-field
theory for confirmaing validity and generality of results.  Skyrme Hartree-
Fock plus BCS pairing (SHFBCS) with SkM∗ interaction [28] was used. The
pairing functional is monopole-type, smoothed constant $G$ model. [29] The
pairing strength is obtained self-consistently by satisfying the nucleon
number conservation and solving the gap equation. The computational code we
use was the one developed in Ref. [30].

##  3 Results and Discussion

###  3.1 Total Energy

In this paper, we focus on the fifteen actinides with the charge number
$Z=92-96$,
${}^{232}{\rm{U}},~{}^{234}{\rm{U}},~{}^{236}{\rm{U}},~{}^{238}{\rm{U}},~{}^{240}{\rm{U}},~{}^{232}{\rm{Pu}},~{}^{234}{\rm{Pu}},~{}^{238}{\rm{Pu}},~{}^{240}{\rm{Pu}},~{}^{242}{\rm{Pu}},~{}^{244}{\rm{Pu}},~{}^{242}{\rm{Cm}},\\\
~{}^{244}{\rm{Cm}},~{}^{246}{\rm{Cm}},~{}^{248}{\rm{Cm}}$, as representative
compound systems of neutron-induced fission reactions to investigate the
triaxiality effects on the height of inner fission barriers.

First, we show in upper panels of Fig. 1 the $\beta_{2}$ dependence of the
total energy of ${}^{240}{\rm{Pu}}$ for axial symmetry (solid line) and
triaxiality (dashed line) cases. Here, $\beta_{2}$ stands for the quadrupole
moment.

Figure 1: (Upper panels)  The $\beta_{2}$ dependence of total energy of
${}^{240}{\rm{Pu}}$. The solid and dashed lines represent axial symmetry and
triaxiality, respectively. The leftmost panel is the result with DD-ME2, the
middle one is that with DD-PC1, and the rightmost one by the SHFBCS model,
respectively.
(Lower panels) The $\gamma$-deformation dependence of ${}^{240}{\rm{Pu}}$ on
$\beta_{2}$. The results from left to right panels were calculated by DD-ME2,
DD-PC1 interactions, and SHFBCS model, respectively.

We found that the global behaviour of the total energy seems to be common in
two RMF calculations and SHFBCS calculation; the ground state appears at
$\beta_{2}\simeq 0.3$ in both assumptions on axial symmetry. On the other
hand, the position of the saddle point depends on the symmetry assumption,
namely, the saddle is at $\beta_{2}\simeq 0.5$ in the axial symmetry cases,
while the saddle is at $\beta_{2}\simeq 0.6$ in the triaxial cases. Note that
we do not performed RMF calculations for $\beta_{0}>0.7$, though there would
exist a second minum like the Skyrme case. In the RMF case, we employed the
harmonic oscillator basis expansion method which has less reliability for
large deformation.

We defined the height of the inner fission barrier as the difference between
the total energies of the saddle point and the ground state. The inner fission
barrier heights obtained from upper panels of Fig. 1 are given in Table 3.

The inner fission barrier height (MeV) of ${}^{240}{\rm{Pu}}$.  Symmetry /
model DD-ME2 DD-PC1 SkM∗ Axial symmetry $9.40$ $8.61$ 10.16 Triaxiality $7.50$
$6.53$ 7.93

From Table 3, we found that the triaxiality makes the inner barrier height
lower than that for the axial symmetry by about 2 MeV, as pointed out by the
original work of DD-PC1 [22]. However, the experimental height of the inner
barrier of ${}^{240}{\rm{Pu}}$ is $5.60~{}{\rm{MeV}}$ [2] or
$6.05~{}{\rm{MeV}}$ [34] and still lower than those theoretical values with
triaxiality listed in Table 3. This discrepancy suggests that theoretial
fission barries must be lowered down further. The SHFBCS calculation has
similar trend. The axail-symmetric calculation overestimates the fission
barrier height by $\sim$ 4 MeV. Taking triaxiality improve this overestimation
by 2 MeV, but still an overestimate remains.  We can observe a similar
discrepancy between experimental inner barrier heights and theoretical heights
assuming triaxiality in other actinides, as will be shown in Fig. 4 later. It
is our experience on the pairing strength as described in the last
publication[16] which helps to reduce the discrepancy. In the following
subsections, we will show the critical effect of the pairing interaction to
lower the inner fission barrier. Before doing so, we wish to discuss detailed
feature of results for ${}^{240}{\rm{Pu}}$ obtained with DD-ME2 and DD-PC1
under both assumptions of axial symmetry and triaxiality.

One of the reasons which brought the difference of the inner fission barrier
heights obtained by the axial symmetry and the triaxial asymmetry is the level
density at the saddle. Figures 2 (with DD-ME2) and 3 (with DD-PC1) depict
proton and neutron single-particle levels of ${}^{240}{\rm{Pu}}$ at the saddle
point under the assumption of axial symmetric deformation and triaxial
deformation. The neutron Fermi energies in 240Pu with DD-ME2 are almost the
same in two cases; $E_{F,n}=-5.85$ MeV for axial symmetric case and
$E_{F,n}=-5.81$ MeV for triaxial case. There are seven levels in the energy
range of $E_{F,n}\pm 1$ MeV for axial symmetric, while five levels lie for
triaxial cases. Similar trends are seen in proton levels and also in those
with DD-PC1 (Fig. 3). For protons, the levels around the Fermi energy are
sparser for the triaxial case than in the axial case. We can see that the
triaxial deformation yields a smaller level density than the axial symmetry
deformation at the Fermi energy.

Figure 2: Neutron (left) and Proton (right) single-particle energies of
${}^{240}{\rm{Pu}}$ at saddle point assuming axial symmetry and triaxiality
calculated with DD-ME2. The horizontal dashed linse denote the Fermi energy
(or chemical energy in BCS theory).  Figure 3: Neutron (left) and Proton
(right) single-particle energies of ${}^{240}{\rm{Pu}}$ at saddle point for
axial symmetry and triaxiality calculated with DD-PC1. The horizontal dashed
linse denote the Fermi energy (or chemical energy in BCS theory).

Lower panels of Fig. 1 represents the $\gamma$-deformation of
${}^{240}{\rm{Pu}}$ as a function of $\beta_{2}$. The shape near the ground
state is axially symmetric, while that near the saddle point
($0.35<\beta_{2}<0.6$) is triaxial. It is known that the $\gamma$-deformation
near the saddle point is almost universal in actinides [35] at $\gamma\simeq
10^{\circ}$.  Another noticeable feature of the relation between the
$\gamma$-deformation and $\beta_{2}$ is the sudden jump around
$\beta_{2}=0.3$. When $\beta_{2}$ is fixed, there are two local minuma at
$\gamma=0$ and $\gamma\simeq 10^{\circ}$. These two states have close
energies, and the lower-energy state chages at $\beta_{2}\sim 0.3$. This
transition causes the discontinuity of $\gamma$ deformation. Similar
discontinuities in multidimensional potential energy surfaces of actinides
nuclei are reported [31, 32, 33].

###  3.2 Effect of triaxiality and pairing interaction for inner fission
barrier

The previous subsection picked out ${}^{240}{\rm{Pu}}$ as a representative
case and showed various features with axial-symmetry/triaxiality. In the
former part of this subsection, we display the inner fission barriers of
actinides, including ${}^{240}{\rm{Pu}}$, as a function of the mass of a
fissioning nuclei. It suggests that we have to reduce the calculated fission
barriers, even if we consider the triaxiality, to reproduce the experimental
data not only in ${}^{240}{\rm{Pu}}$ but also in other actinides. Then, in the
latter part of this subsection, we focus on the influence of the pairing
interaction on the fission barrier heights. Our previous study for the axial-
symmetric cases showed that it could lower the heights of inner fission
barriers .

Figure 4 compares the experimental inner-fission-barrier heights [2, 34] of
actinides with those using two interactions (DD-ME2 and DD-PC1) in axial-
symmetric and triaxial cases. These results show that the inner fission
barrier is generally about $0.5\sim 5$ MeV lower than the axial-symmetric case
when we impose the triaxiality. We also found that the reduction width depends
on the number of neutrons and protons. However, the barrier heights of a few
nuclei (e.g. ${}^{240}{\rm{Pu}}$) are still overestimated by about 2 MeV
compared to the experimental values, even assuming the triaxiality.

Figure 4: Comparison of the calculation results of inner fission barrier by
axial symmetry and triaxiality with experiment values for Pu isitopes[2, 34].

The above problem of overestimating experimental values can be solved by
increasing the pairing strength. Here, we define the pairing strength as a
single multiplier parameter $\alpha$ such as $G\rightarrow\alpha G$. We show
the results of inner fission barriers with unchanged pairing strength
($\alpha=1.00$) and with changed pairing strength by $+20\%$ ($\alpha=1.20$)
in Fig. 5. Here, experimental values $B_{f}(\mbox{Exp})$ were taken from [2,
34]. In addition, as an example, we selected three Pu-isotopes, where the
fission barriers with triaxiality still deviate from the experimental values
by more than $1$ MeV. Such a combination of triaxiality and pairing
interaction, both models (DD-ME2 and DD-PC1) can drastically improve the
reproducibility of the experimental fission barriers in Pu-isotopes by
increasing the pairing strength regardless of axial symmetry (blue unfilled
circles) or triaxiality (red unfilled squares) as shown in Fig. 5.

Thus, one of the reasons why the height of the inner fission barrier changes
depending on the strength of the pairing force is because the single-particle
level densities around the Fermi surface at the saddle point is higher than
the ground state. It leads to the easier occurrence of interactions between
particles at the saddle than at the ground state. Thus, the particles at the
saddle can gain larger paring energy to the negative direction. Then stronger
pairing interaction can reduce the fission barrier height. For a more detailed
discussion and analysis, see the reference [36]. In the referred article, the
authors mentioned the case only with axial symmetry; however, there is no
significant difference in a physical picture between axial symmetry and
triaxiality.

The primary effect of the pairing interaction on the fission barrier, i.e.
reduction of the barrier height, is common in both symmetries. However, the
strength of that effect depends on the symmetry, axial-symmetry or
triaxiality, as shown in Fig. 5. The $B_{f}$ change due to the pairing
strength $\alpha$ is larger in the axial-symmetric case than in the triaxial
case because the level density assuming axial-symmetry is denser around the
Fermi energy than in the triaxial case. It suggests that the sensitivity of
the $B_{f}$ on $\alpha$ comes from the difference in the level density
structures between these two symmetries.

The above discussion on Pu-isotopes concludes that the larger pairing strength
can lower the inner fission barrier. We can expect that such a combination of
the triaxiality and the pairing interaction would effectively reduce the
fission barrier height in the other actinides. However, the best parameter
search on $\alpha$ requires a very high computational cost, which will be our
following paper’s subject.

Figure 5: The changes of inner fission barrier by each pair strength on axial
symmetry and triaxiality. The result of the left panel was calculated by DD-
ME2, the right one was calculated by DD-PC1.

Finally, we refer to why we set the pairing strength of Pu-isotope as
$\alpha$=1.2. We adjusted $\alpha$ to reproduce the experimental value of the
pairing rotational energy. Eq.(9) states that the pairing rotational energy
depends only on the ground state energy, which is generally the same in axial-
symmetric and triaxial cases, as shown in Fig. 1, i.e., $\gamma=0$ at the
ground state. It means that the pairing rotational energy with the triaxiality
is the same as that with axial symmetry. Figure 6 shows the resulting pairing
rotational energy with DD-PC1 (uppermost panels) and DD-ME2 (middle panels) in
the Pu isotopes. These results reveal that making pairing strength stronger by
$+20\%$ (blue filled triangles) is the most reasonable among three cases
($0.8G,G,1.2G$) to explain the experimental value (white unfilled squares) of
pairing rotational energy. The SHFBCS calulations (lowermost panels) also show
that the pairing strength with $1.2G$ repoduces the pairing rotational
energies better than the others, while its differeces are small. Therefore, we
can notice herre that the effect of the pairing on the nuclear properties has
a strong model dependence. Similar results are obtained in Uranium isotopes.
Therefore, we can draw an important conclusion that the pairing strengths in
the actinide region are not subject to those determined by analysis for
lighter nuclei, but rather, they should be determined systematically by taking
account of the nuclear binding energy, fission barrier and pairing rotational
energy simultaneously. For this aim, we need to consider both of the triaxial
deformation and pairing interaction in a consistent manner.

Figure 6: The changes of pairing rotational energy by each pair strength. The
result was calculated by DD-ME2 (upper), DD-PC1 (middle), and SkM∗ (lower).

##  4 Summary

We systematically investigated the height of the inner fission barrier of
actinide nuclei based on the density-dependence relativistic mean-field theory
(DD-ME2 and DD-PC1), with taking account of the triaxial deformation. We found
that in both the DD-ME2 and DD-PC1 parameter sets, triaxial deformation could
reduce the inner fission barrier height compared to ones in axial symmetric
deformed cases by about $0.5\sim 5$ MeV. However, we also found that the
fission barrier height with triaxiality still deviates from the experimental
value in some nuclides, for instance, $A\geq 238$. For such nuclides, we
increased the pairing strength by $20\%$ to improve reproducibility of the
barrier height. That strength also provides more realistic pairing rotational
energy, which is an observable sensitive to the pairing correlation. Thus, we
obtained a consistent picture of the fission barrier and the pairing
rotational energy. We obtained similar results with use of non-relativistic
theory, Skyrme-HF plus BCS pairing. The important conclusion we could draw
from the present work is that the pairing strengths in the actinide region are
not subject to those determined for lighter nuclei, but rather, they should be
determined systematically by taking account of the nuclear binding energy,
fission barrier and pairing rotational energy simultaneously. For this aim, we
need to consider both of the triaxial deformation and pairing interaction in a
consistent manner.

## Acknowledgements

This study is supported by Grands-in-Aid for Scientific Research (B) No. 21H
01856 from Japan Society for the Promotion of Science.

## References

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