Also at ]Kazan Federal University, Kazan 420008, Russia

# Entropies, level-density parameters and fission probabilities along the
triaxially- and axially-symmetric fission paths in 296Lv

A. Rahmatinejad  Joint Institute for Nuclear Research, Dubna, 141980, Russia
T. M. Shneidman  [  Joint Institute for Nuclear Research, Dubna, 141980,
Russia  G. G. Adamian  Joint Institute for Nuclear Research, Dubna, 141980,
Russia  N. V. Antonenko  Joint Institute for Nuclear Research, Dubna, 141980,
Russia  P. Jachimowicz  Institute of Physics, University of Zielona Góra,
Szafrana 4a, 65516 Zielona Góra, Poland  M. Kowal<EMAIL_ADDRESS>National Centre for Nuclear Research, Pasteura 7, 02-093 Warsaw, Poland

###### Abstract

We employ a statistical approach to investigate the influence of axial
asymmetry on the nuclear level density and entropy along the fission pathways
of a superheavy nucleus, specifically focusing on the 296Lv isotope. These
pathways are determined within multidimensional deformation spaces. Our
analysis reveals a significant impact of triaxiality on entropy. Additionally,
suppressing of shell effects can alter the fission scenario depending on the
available excitation energy. We derive the deformation-dependent level density
parameter which plays a crucial role in estimating the survival probability of
a superheavy nucleus. Furthermore, we utilize a set of master equations to
obtain the time-dependent fission probabilities and calculate the ratio of
decay probabilities for both axial and triaxial paths

###### pacs:

21.10.Ma, 21.10.Pc, 24.60.Dr, 24.75.+i
Keywords: microscopic-macroscopic model, fission barrier, level-density
parameter, survival probability, superheavy nuclei

##  I Introduction

Recent breakthroughs in synthesizing novel superheavy nuclei, as documented in
the works of Oganessian et al. Og1 ; Oganessian1 ; Oganessian2 , have resulted
in substantial advancements. Building upon these achievements, forthcoming
experiments are planned to extend the frontiers of heavy element production,
capitalizing on the cutting-edge ”superheavy factory” facility recently
established at JINR (Dubna).

The successful production of superheavy nuclei relies heavily on their ability
to resist fission. This resistance is crucial in determining the survival
probability of a hot compound nucleus formed in complete fusion reactions. The
competition between neutron emission and fission plays a significant role in
shaping the outcome of the decay process and it depends on factors such as the
level densities and the potential energy surface (PES) topology. These
characteristics mainly predestine the formation of evaporation residue nuclei.
The role of entropy in this process is less studied and was not frequently
taken up in studies. The critical advantage of entropy is its ability to
encode information about the density of states across diverse potential energy
landscapes simultaneously. However, the concept of entropy becomes more
nuanced and challenging to define precisely. This is primarily due to the
following reasons:

  * •

Finite Size: Unlike macroscopic systems, which typically consist of many
particles, heavy nuclei are relatively small and finite systems. As a result,
the statistical behavior and thermodynamic quantities, including entropy,
might exhibit deviations from the behavior predicted by classical statistical
mechanics.

  * •

Quantum Mechanical Nature: Nuclei are governed by quantum mechanics which
introduces inherent uncertainties and restrictions on the states available to
individual particles. The discrete energy levels and quantum correlations make
entropy more intricate than that of the classical systems.

  * •

Lack of Equilibrium: Determination of the entropy in a nucleus requires
achieving thermal equilibrium, which is challenging for isolated individual
nuclei. Unlike macroscopic systems that readily reach equilibrium through
interactions with a heat bath, individual nuclei are not typically in thermal
contact with a large reservoir, making less straightforward the application of
equilibrium statistical mechanics.

Nevertheless, various statistical approaches and theoretical models have been
developed to estimate and describe the entropy of heavy nuclei. These methods
incorporate the concepts of nuclear level densities, statistical ensembles,
and thermodynamic considerations to provide insights into the statistical
behavior and thermodynamic properties of nuclear systems. So, the concept of
entropy is still in use.

Calculating level densities, we should determine the eigenvalues and their
degeneracy of the nuclear Hamiltonian and to count the number of states within
a specific energy interval of interest. Due to the exponential increase in the
total number of states with excitation energy above a few MeV, statistical
methods are employed to tackle the problem. Several sophisticated
combinatorial methods, such as those described in Refs. hilary ; hilary2 ,
have been developed, where factors like parity, angular momentum, pairing
correlations, and collective enhancements are explicitly accounted for using
the Gogny interaction. Additionally, a general and exact scheme for
calculating particle-hole level densities while considering the Pauli
exclusion principle is presented in Ref. Blin . The role of microscopic level
densities in the fission process and their influence on nuclear shape
evolution with excitation energy are discussed in Refs. Aberg1Aberg3Aberg4 .
Relativistic mean-field theory is employed to calculate nuclear level
densities in Ref. pomorska1 . In contrast, a self-consistent mean-field
approach is utilized to determine single-particle level densities at various
temperatures and the level density parameters are calculated using the Yukawa-
folded potential in Ref. pomorska2pomorska3 . Spin- and parity-dependent
nuclear level densities obtained with the moment method in the proton-neutron
formalism are presented in Ref. Senkov . Finally, a direct microscopic
calculation of nuclear level densities with the shell model Monte-Carlo
approach is discussed in Ref. Alhassid2015 . In practical applications,
certain approximations and assumptions are still necessary, including
corrections for superfluidity effects and collective rotational and
vibrational enhancements.

The well-established Fermi-gas model can effectively describe the transition
from a paired system to a system of non-interacting fermions as the nuclear
system goes from low to higher energies. This phenomenological model accounts
for the pairing effect by introducing a constant back-shift parameter $\Delta$
of the excitation energy. In the Fermi-gas model, the average level-density
parameter, which relates the excitation energy to the nuclear temperature, is
often assumed to have a linear dependence on the mass number $A$ Sokolov1990 .
However, the level-density parameter exhibits also energy dependence and
gradually approaches an asymptotic value as the energy increases beyond the
neutron separation energy. To incorporate the energy and shell correction
dependencies into the level density parameter, a phenomenological expression
was introduced in Ref. Ignatyuk1975 .

Our recent findings Rahmatinejad2021  have demonstrated that the level density
parameter $a$, which plays a crucial role in estimating survival
probabilities, strongly depends on the available excitation energy,
particularly at low energies. Given its appearance as an exponential function
in the decay rates, even minor variations in this parameter can profoundly
impact the estimated survival probabilities of superheavy nuclei. In this
study, we will derive this parameter as a function of deformation, providing a
more accurate and comprehensive description.

The primary objective of this article is to combine advanced techniques,
including the multidimensional manifold of deformations for fission
trajectories, with a statistical approach for computing nuclear level density
and entropy, and to finally calculate the level density parameter along
fission paths. We discuss the calculations of two extreme cases, axial and
non-axial fission pathways for the 296Lv nucleus.

It is important to note that the level density at any point along specific
fission paths is not the same as at the ground state (GS). This is primarily
due to two factors. Firstly, the excitation energy available at any point of
the fission paths is reduced by the difference in deformation energy between
this point and the GS. Secondly, the distribution of single-particle levels is
altered due to the change of the shape of compound nucleus. Thus, identifying
the correct paths that govern the fission process is essential.

##  II  Method of Calculation

We adopt a two-step methodology in our study. Firstly, we construct a
5-dimensional PES using the macroscopic-microscopic (MM) method. This PES
allows us to identify suitable axial and non-axial fission paths by
considering the direction with the most substantial gradient drop in all
deformation variables. In the second step, we employ a statistical approach to
determine the density of states. Based on this information, we calculate the
entropy and relevant parameters along the fission paths.

###  II.1  Potential Energy Surface

The PES are calculated using MM method. In this approach, the microscopic
energy is determined by employing the Strutinski shell and pairing correction
method STRUT67 , which involves diagonalization of the deformed Woods-Saxon
potential WS  to obtain single-particle levels. We consider $n_{p}=450$ lowest
proton levels and $n_{n}=550$ lowest neutron levels from the $N_{max}=19$
lowest shells of the harmonic oscillator in the diagonalization process. The
standard values of $\hbar\omega_{0}=41/A^{1/3}$ MeV for the oscillator energy
and $\gamma=1.2\hbar\omega_{0}$ for the Strutinski smearing parameter are
used. Additionally, a six-order correction polynomial is applied to calculate
the shell correction.

For the macroscopic component, we employ the Yukawa plus exponential model KN
with parameters specified in Ref. WSpar . The deformation-dependent Coulomb
and surface energies are integrated using a 64-point Gaussian quadrature
method.

In order to construct the PES, a five-dimensional deformation space is
utilized, which includes an expansion of the nuclear radius:

$\displaystyle R(\vartheta,\varphi)=cR_{0}\\{1$ $\displaystyle+$
$\displaystyle\beta_{20}{\rm Y}_{20}+{\beta_{22}\over{\sqrt{2}}}[{\rm
Y}_{22}+{\rm Y}_{2-2}]$ (1) $\displaystyle+$ $\displaystyle\beta_{40}{\rm
Y}_{40}+\beta_{60}{\rm Y}_{60}+\beta_{80}{\rm Y}_{80}\\},$

where the quadrupole non-axial deformation parameter $\beta_{22}$ is
explicitly included. For each nucleus, we generate the following 5D grid of
deformations:

$\displaystyle\beta_{20}$ $\displaystyle=$
$\displaystyle\phantom{-}0.00\;(0.05)\;0.60,$ $\displaystyle\beta_{22}$
$\displaystyle=$ $\displaystyle\phantom{-}0.00\;(0.05)\;0.45,$
$\displaystyle\beta_{40}$ $\displaystyle=$ $\displaystyle-0.20\;(0.05)\;0.20,$
(2) $\displaystyle\beta_{60}$ $\displaystyle=$
$\displaystyle-0.10\;(0.05)\;0.10,$ $\displaystyle\beta_{80}$ $\displaystyle=$
$\displaystyle-0.10\;(0.05)\;0.10.$

A grid of $29,250$ points (nuclear shapes) is employed, with the grid steps
specified in parentheses. This primary grid, labeled as (II.1), is extended
through fivefold interpolating in all directions. As a result, an interpolated
energy grid consisting of over 50 million points is obtained (see Refs.
Jachimowicz2017_II ; Jachimowicz2017_III  for more details). All the
parameters are specified in the last tables of Ref. table2021 .

###  II.2  Level-density parameters

Based on the superfluid formalism Decowski1968 ; origin1  and an assumption of
thermal equilibrium between neutron and proton subsystems, the excitation
energies $(U=U_{Z}+U_{N})$, entropies $(S=S_{Z}+S_{N})$ and intrinsic level
densities $\rho$ are calculated at each temperature $T$ as

$E_{N(Z)}(T)=2\sum_{k}\varepsilon_{k}n^{N(Z)}_{\Delta,k}-\frac{\Delta^{2}_{N(Z)}}{G_{N(Z)}},$
(3) $U_{N(Z)}(T)=E_{N(Z)}(T)-E_{N(Z)}(0),$ (4)

$\displaystyle
S_{N(Z)}(T)=\sum_{k}\left\\{\ln\left[1+\exp{\left(-\frac{E^{N(Z)}_{k}}{T}\right)}\right]\right.$
$\displaystyle\left.+\frac{E^{N(Z)}_{k}}{T\left[1+\exp{\left(\frac{E^{N(Z)}_{k}}{T}\right)}\right]}\right\\},$
(5)

$\rho=\frac{\exp{(S)}}{(2\pi)^{\frac{3}{2}}\sqrt{D}}.$ (6)

The determinant of the matrix composed of the second derivatives of the
entropy with respect to $1/T$ and $\mu=\lambda/T$ is denoted as $D$. In the
equations above, the occupation probabilities

$n_{\Delta,k}^{N(Z)}(T)=\frac{1}{2}\left(1-\frac{\varepsilon^{N(Z)}_{k}-\lambda_{N}}{E^{N(Z)}_{k}}\tanh\frac{E^{N(Z)}_{k}}{2T}\right)$
(7)

and the quasiparticle energies
$E^{N(Z)}_{k}=\sqrt{(\varepsilon^{N(Z)}_{k}-\lambda_{N(Z)})^{2}+\Delta_{N(Z)}^{2}}$
are calculated using the single-particle energies ($\varepsilon^{N(Z)}_{k}$)
of the Woods-Saxon potential. The constants of the pairing interaction for
neutrons ($G_{N}$) and protons ($G_{Z}$) are adjusted to obtain correct GS
pairing gaps ($\Delta_{N}$ and $\Delta_{Z}$) in the MM method with using the
Bardeen-Cooper-Schrieffer (BCS) equations:

$N=2\sum_{k}n^{N}_{\Delta,k}(T),$ (8)
$\frac{2}{G_{N}}=\sum_{k}\frac{1}{E^{N}_{k}}\tanh\frac{E^{N}_{k}}{2T}$ (9)

for neutrons and

$Z=2\sum_{k}n^{Z}_{\Delta,k}(T),$ (10)
$\frac{2}{G_{Z}}=\sum_{k}\frac{1}{E^{Z}_{k}}\tanh\frac{E^{Z}_{k}}{2T}$ (11)

for protons at zero temperature. Using the values of pairing constants
obtained, the pairing gaps $\Delta_{N(Z)}$ and chemical potentials
$\lambda_{N(Z)}$ are determined by solving Eqs.(8)–(11) at given temperatures.
We repeated the calculations using the single-particle level energies obtained
with the Woods-Saxon potential at each given set of deformations
($\beta=\beta_{20},\beta_{22}$) along the fission path.

Above the critical temperature ($T_{cr}$), the pairing gap disappears, and all
thermodynamic quantities revert to those of a noninteracting Fermi system.
Generally, a larger density of states close to the Fermi surface at the saddle
point (SP) leads to a larger pairing correlations and, as a result, to a
larger critical temperature compared to the GS. In our calculations in the
superheavy mass region, the critical temperatures for neutrons and protons are
about $0.42$ MeV at the GS and $0.52$ MeV at the SP. The corresponding total
excitation energies are approximately $U_{cr}\approx 5.14$ MeV at the GS and
$U_{cr}\approx 11.27$ MeV at the SP.

Fitting the calculated values of the intrinsic level density with the back-
shifted Fermi gas expression

$\rho_{FG}(U)=\frac{\sqrt{\pi}}{12a^{\frac{1}{4}}(U-\Delta)^{\frac{5}{4}}}\exp({2\sqrt{a(U-\Delta)}})$
(12)

we obtain the level density parameter $a(U)$ as a function of excitation
energy. In the calculations, the energy back-shifts are taken as
$\Delta=12/\sqrt{A}$, 0, and $-12/\sqrt{A}$ MeV for even-even, odd, and odd-
odd isotopes, respectively.

At finite temperatures, the microscopic corrections to energy are replaced by
microscopic corrections to the free energy
$\delta{F}(T)=\delta{F}_{shell}(T)+\delta{F}_{pair}(T)$. Hence, the pairing
and shell corrections to entropy, as well as the energy, are required at each
temperature. In general, the temperature-dependent energy and entropy are
defined as

$E(T)=2\int_{-\infty}^{\infty}n(T,\lambda)\varepsilon
g(\varepsilon)d\varepsilon$ (13)

and

$\displaystyle
S(T)=-\int_{-\infty}^{+\infty}[(1-n(T,\lambda))\log(1-n(T,\lambda))+$
$\displaystyle n(T,\lambda)\log n(T,\lambda)]g(\varepsilon)d\varepsilon,$ (14)

respectively. Here, $g(\varepsilon)$ is the density of the single-particle
levels and temperature-dependent occupation probability is taken as
$n(T,\lambda)=1/(1+e^{(\varepsilon-\lambda)/T})$ Ivanyuk2018 .

To obtain the shell correction at finite temperature
$\delta{F}_{shell}=E(T)-\tilde{E}(T)-T\left[S(T)-\tilde{S}(T)\right]$, we
calculate the energy $E(T)$ and entropy $S(T)$ for the discrete spectrum by
taking $g(\varepsilon)=\sum_{k}\delta(\varepsilon-\varepsilon_{k})$ in
(13,II.2). The energy $\tilde{E}(T)$ and entropy $\tilde{S}(T)$ for the
smoothed spectrum are calculated with occupation number taken as
$n(T,\tilde{\lambda})=1/(1+e^{(\varepsilon-\tilde{\lambda})/T})$ and the
smooth density $\tilde{g}(\varepsilon)$ of the single-particle levels as

$\tilde{g}(\varepsilon)=\frac{1}{\gamma\sqrt{\pi}}\sum_{k=1}e^{-x^{2}}\sum_{n=0,2,...}^{6}c_{n}H_{n}(x),$
(15)

where $x=(\varepsilon-\varepsilon_{k})/\gamma$, $c_{0}=1$, and
$c_{n+2}=-c_{n}/(n+2)$ STRUT67 . The chemical potentials
($\lambda,\tilde{\lambda}$) are defined by the particle number conservation
condition

$N(Z)=2\int_{-\infty}^{+\infty}n(T,\lambda)g(\varepsilon)=2\int_{-\infty}^{+\infty}n(T,\tilde{\lambda})\tilde{g}(\varepsilon)d\varepsilon.$
(16)

The temperature-dependent pairing correction is calculated as
$\delta{F}_{pair}=E_{pair}-\tilde{E}_{pair}-T\left[S_{pair}-\tilde{S}_{pair}\right]$.
The pairing energy, $E_{pair}(T)=E^{BCS}(T)-E^{BCS}_{\Delta=0}(T)$,
corresponding to the real single-particle level distribution is calculated
using

$E^{BCS}_{N(Z)}(T)=2\sum_{k}\varepsilon_{k}n^{N(Z)}_{\Delta,k}-\frac{\Delta^{2}_{N(Z)}}{G_{N(Z)}}-G\sum_{k}\left(n_{\Delta,k}^{N(Z)}\right)^{2},$
(17)

where $n_{\Delta,k}^{N(Z)}$ is defined in Eq. (7). Note, that the last term of
(17) is often neglected in the calculation of excitation energy under the
standard assumption of the BCS model Bolsterli1972 . However, it should be
taken into account in the microscopic corrections because of the different
pairing constants used in the calculations for the discrete and smoothed
spectra.

The pairing energy corresponding to the smooth single-particle level
distribution
$\tilde{E}_{pair}(T)=\tilde{E}^{BCS}(T)-\tilde{E}_{\Delta=0}^{BCS}(T)$ is
calculated using

$\displaystyle\tilde{E}^{BCS}_{N(Z)}(T)=2\int_{-\infty}^{+\infty}\varepsilon\tilde{g}(\varepsilon)\tilde{n}(T)d\varepsilon-\frac{\tilde{\Delta}_{N(Z)}^{2}}{\tilde{G}_{N(Z)}}$
$\displaystyle-\tilde{G}\int_{-\infty}^{+\infty}\tilde{g}(\varepsilon)\tilde{n}^{2}(T)d\varepsilon.$
(18)

Here,

$\displaystyle\tilde{n}(T)=\frac{1}{2}\left(1-\frac{\varepsilon-\tilde{\lambda}_{N(Z)}}{\tilde{E}_{N(Z)}}\tanh\frac{\beta\tilde{E}^{N(Z)}}{2}\right)$
(19)

is the occupation probability and
$\tilde{E}^{N(Z)}=\sqrt{(\varepsilon-\tilde{\lambda}_{N(Z)})^{2}+\tilde{\Delta}_{Z(N)}^{2}}$
is the quasiparticle energy. The chemical potentials $\tilde{\lambda}_{Z(N)}$
and pairing gaps $\tilde{\Delta}_{N(Z)}$ for smoothed spectrum are calculated
by solving analogs of BCS equations:

$\frac{2}{\tilde{G}_{N(Z)}}=\int_{-\infty}^{+\infty}\frac{1}{\tilde{E}^{N(Z)}}\tanh\left(\frac{\beta\tilde{E}^{N(Z)}}{2}\right)\tilde{g}(\varepsilon)d\varepsilon,$
(20)
$N(Z)=\int_{-\infty}^{+\infty}\left(1-\frac{\varepsilon-\tilde{\lambda}_{N(Z)}}{\tilde{E}^{Z(N)}}\tanh\frac{\beta\tilde{E}^{N(Z)}}{2}\right)\tilde{g}(\varepsilon)d\varepsilon.$
(21)

The pairing constant in Eq. (20) is adjusted to reproduce the smoothed pairing
energy at $T=0$ which is obtained with the assumption of a constant density of
pairs near the Fermi level table2021 .

The pairing correction to entropy is calculated in a similar way. The average
entropy $\tilde{S}_{N(Z)}^{BCS}(T)$ is obtained by replacing the sum over
discrete states $k$ in Eq.(II.2) by integrals with the average density of
states defined in Eq. (15) and replacing the quasiparticle energy
$E^{N(Z)}_{k}$ by its smooth analog $\tilde{E}^{N(Z)}$.

##  III  Results and discussion

In Fig. 1, we present the PES of 296Lv projected onto the
$(\beta_{20},\beta_{22})$ plane. Among the various trajectories, we selected
two for in-depth analysis. The first trajectory, marked in red, encompasses
the triaxial shapes, while the second trajectory, depicted in blue,
corresponds to the axial shape. The energy landscape is determined by
minimizing the energy on a five-dimensional grid (II.1) concerning
$\beta_{40}$, $\beta_{60}$, and $\beta_{80}$. The inclusion of quadrupole
nonaxial deformation $\beta_{22}$ in (1) significantly modifies the landscape
and plays a crucial role in the depiction of the first SP. As depicted on the
map, the impact of this effect is evident in the substantial reduction of the
axial barrier by approximately 1 MeV. However, it is crucial to consider that
mapping of energy in a multidimensional space poses some challenges. When
minimizing certain deformations, the dimensionality reduction often results in
a PES composed of disconnected patches corresponding to multiple minima in the
discarded dimensions. Hence, the actual SP is identified through the imaginary
water flow technique in the total deformation space.

Figure 1: The potential energy landscape of 296Lv is projected onto the
$(\beta_{20},\beta_{22})$ plane. The dashed line represents the potential
fission path from the GS to the SP. The energy scale is measured in MeV and is
calculated relative to the macroscopic energy at the spherical shape.

In the initial step, various thermodynamic quantities such as entropy,
intrinsic level density, and energy-dependent shell corrections are computed
as functions of excitation energy for each deformation along the selected
fission path. Neglecting the kinetic energy for the motion in deformation
space for each set of deformation $\beta=(\beta_{20},\beta_{22})$ we have

$U(\beta)+E_{pot}(\beta)=E_{0}\equiv\text{constant},$ (22)

where $E_{pot}(\beta)=E_{mac}(\beta)+E_{mic}(\beta)$ is the corresponding
potential energy and $U(\beta)$ is the excitation energy. If damping of
microscopic correction with excitation energy is taken into account, the Eq.
(22) should be replaced with

$U^{*}(\beta)+E_{pot}(\beta,U)=E_{0},$ (23)

where $E_{pot}(\beta,U)=E_{mac}(\beta)+E_{mic}(\beta,U^{*})$ represents the
damped potential energy. Calculating the level density and entropy at
$U^{*}(\beta)$, we consistently take into account the excitation energy-
dependent potential as well as the change of thermodynamic quantities due to
variation of excitation energy with respect to the damped potential.

The properties of the nucleus at various deformations are calculated at the
same value of total energy $E_{0}$ or, equivalently, at the given excitation
energy of the GS: $U_{0}=E_{0}-E_{mac}(GS)-E_{mic}(GS)$. The energy-dependent
microscopic part of the potential $E_{mic}(\beta,U^{*})$ is obtained as
explained in Sec. II.2.

###  III.1 Fission Paths

In our detailed analysis, we will focus on the entropy and level density
parameters along specific fission paths. We have chosen the paths that exhibit
the greatest gradient drop in all variables, and we have determined the
potential to ensure that these paths pass through two uniaxial SP. The axial
splitting path is depicted in blue in Fig. 2. Additionally, the macroscopic
and microscopic components of the potential energy are shown in green and red,
respectively. In Fig. 2(b), we show the same analysis, maintaining the same
color hierarchy, but for the non-axial path. Here, we observe notable
differences with Fig. 2(a). There is essentially no macroscopic barrier along
this path. Instead, it is the shell effect, as expected, that creates the
barrier and stabilizes the system. Furthermore, we can clearly see that the
triaxial barrier is extensive and spread out, whereas the axial barrier is
more compact. The competition between the height and shape of these barriers
becomes crucial when considering the transmission process, especially when the
available energy is lower than the fission barrier (tunneling).

Figure 2: The variation of potential energy, macroscopic energy, and
microscopic correction at zero excitation energy is depicted as a the function
of the deformation parameter $\beta_{2}$ for axial (a) and triaxial (b) paths
of fission of 296Lv.

The influence of excitation energy on the GS and the SP is evident as shown in
Fig. 3. The GS represents the lowest energy configuration, characterized by a
well-defined shell structure. As the excitation energy increases, it
introduces additional energy into the system, potentially destabilizing the
shell structure. On the other hand, the SP corresponds to a highly deformed
configuration, where the competition between shell effects and the deformation
energy is crucial. The excitation energy has a more indirect impact on the SP,
primarily affecting the overall energy of the system rather than the specific
internal structure. Nevertheless, the destructive nature of excitation energy
on the entire fission barrier is observed. At higher excitation energies, the
barrier gradually diminishes, as the influence of shell effects becomes less
pronounced, and the barrier approaches the macroscopic energy. Eventually, the
threshold for fission disappears as the increasing excitation energy washes
out the barrier. Also, with increasing $U_{0}$, the potential energy as a
function of $\beta_{2}$ becomes flat near the GS.

In Fig. 4, we present the variation of fission barriers $E_{pot}-E_{pot}^{GS}$
with excitation energy for both axial (solid lines) and triaxial (dashed
lines) fission paths. Note that as stated before, $U_{0}$ is the excitation
energy with respect to cold GS. The actual excitation energy of the GS is
smaller than $U_{0}$ because of the damping effects. For instance, at
$U_{0}=25$ MeV the GS excitation energy with respect to damped potential is
$U^{*}(GS)=20.93$ MeV. The damping effect reduces fission barrier
corresponding to triaxial(axial) path from $B_{f}^{tri(ax)}=6.25(7.31)$ MeV
for cold nucleus to $B_{f}^{tri(ax)}=3.05(3.93)$ MeV at $U_{0}=25$ MeV. Also,
the damping effect shifts the triaxial SP from $\beta_{2}=0.37$ to
$\beta_{2}=0.33$, and the axial SP from $\beta_{2}=0.25$ to $\beta_{2}=0.22$.

Figure 3: The variation of potential energy along axial (a) and triaxial (b)
fission path at $U_{0}=0,15,35$ MeV, together with the macroscopic energy.
Here, $U_{0}$ is the excitation energy calculated concerning the GS without
the damping of shell correction. Figure 4: The variation of fission barriers
with excitation energy $U_{0}=0,15,35$ MeV (a-c), for axial (solid lines) and
triaxial (dotted lines) fission path.

###  III.2 Level-density parameter

In this work, we explore variation of level-density parameter with deformation
along the fission path. The level-density parameter at the GS corresponds to
particle emission channels such as neutron emission ($a_{n}$), while its value
at the SP corresponds to fission ($a_{f}$). It is widely recognized that even
the most minor change in the values of $a_{n}$ and $a_{f}$ can significantly
impact the accuracy of calculations of the nuclear survival probability. Our
investigation encompasses the entire cleavage path, both axially (Fig. 5(a))
and non-axially (Fig. 5(b)) symmetric at $U_{0}=15$ and $35$ MeV. Evidently,
the choice of pathway significantly affects the behavior of the level-density
parameter. The variability is more pronounced along the axial path and appears
smoother along the non-axial path. Additionally, the results demonstrate that
the intricate topology of the PES has a more significant impact on the
parameter at lower energies than anticipated.

As evident from Fig. 5 (see dotted lines), the damping of microscopic
correction systematically influences both fission pathways. Although, this
effect may seem to be small, it is not negligible because it affects the GS
and the SP differently. Therefore, the damping has significant impact on
$a_{f}/a_{n}$ ratio. It is interesting to note that at small $U_{0}$ the value
of $a(SP)$ is comparable or even smaller than $a(GS))$. This is in line with
the conclusion of Ref. Azam_ratios_2022  that the ratio $a_{f}/a_{n}$ exhibits
an increasing trend with excitation energy approaching its asymptotic value
less than 1.1.

Figure 5: (a): Level-density parameters obtained for axial fission path at
$U_{0}=15,35$ MeV with (dotted lines) and without (solid lines) consideration
of damping of microscopical effects. (b): The same as panel (a), but for
triaxial fission path.

To highlight damping effect more prominently, the ratio of level-density
parameters calculated with and without damping along axial (solid line) and
triaxial (dotted line) fission paths is presented in Fig. 6. The calculations
are performed at $U_{0}=30$ MeV. The most significant impact occurs near the
GS. This effect can be attributed to the large shell correction and large
excitation energy at the GS. Similar but less pronounced effect is observed
for the axial path at deformed minimum around $\beta_{20}\sim 0.4$. It is
noteworthy that the observed effect of damping never exceeds 5$\%$.

Figure 6: The ratio of level-density parameters obtained with account for
damping of potential to those obtained without account for damping of
potential along axial (solid line) and triaxial (dashed line) fission path at
$U_{0}=30$ MeV.

###  III.3 Entropy

Figure 7 displays two panels illustrating the variations in entropy along the
axial and non-axial fission paths at energies $U_{0}=15$ and $35$ MeV. Dotted
lines depict the entropy values calculated with damped potential, while the
solid lines represent the results without accounting for damping. By comparing
these two sets of curves, we can assess the influence of damping on the
entropy along these paths. Notably, regardless of the chosen path, the entropy
systematically decreases when potential damping is included.

Figure 7: Comparison between entropy along axial (blue lines) and triaxial
fission paths (red lines) at excitation energies $U_{0}=15$ and $35$ MeV with
(dotted lines), and without (solid lines) consideration of the damping of
potential.

It is apparent that at low energies the behavior of entropy to large extent is
determined by the intricacies of the potential energy and of the density of
single-particle states along the path. The nucleus undergoes significant
changes of deformation as it moves from the GS to the SPs. Along this paths,
the elongation and stretching of the nucleus lead to increase of the density
of single-particle states and, thus, entropy of the deformed configurations is
enhanced. However, at low energies the impact of excitation energy on entropy
is dominant so that the maximum entropy is observed at the GS. With increase
of excitation energy the trend of entropy curves are still defined by the
shape of potential energy, but it is less pronounced because of the growing
role of the density of single-particle states.

The account for the damping, completely changes the behavior of entropy. The
damping of microscopic correction suppresses the role of excitation energy in
entropy but leaves the role of density of single-particle states untouched. As
seen from Fig. 3, because of the significant damping of microscopic
corrections at the GS, the excitation energy of the GS decreases more in
comparison with the deformed configurations. Therefore, the gain in entropy of
the GS due to the larger excitation energy is reduced. This leads to smearing
of the entropy as a function of deformation, irrespective of whether
triaxiality is taken into account or not.

At larger $U_{0}$, due to the significant reduction of the excitation energy
for the configurations around the GS together with the larger density of
states for deformed configurations, the entropy curve gets flattened around
the GS. Moreover, the entropy at the GS becomes smaller than entropy of SPs.

It is worth noting that the difference between entropies at the axial and
triaxial SPs decreases independent of whether or not the damping effect is
taken into account.

In Fig.7, both energy and deformation effects contribute to the displayed
entropy curves. To elucidate the effect of deformation on entropy, we
performed the calculations with fixed excitation energy along the path. Note
that the assumption of constant excitation energy is not realistic, and we do
it only to study the effect of deformation on entropy. In Fig. 8, the
excitation energies are fixed as $U=15$ and $35$ MeV with respect to undamped
potentials (solid lines). Panel (a) shows the ratio of entropies calculated at
each deformation along the axial fission path to those at the GS. Panel (b)
shows the same as panel (a), but for the triaxial path. The trend of these
curves clearly resembles the shape of the potential energy. The effect of
structure gradually decreases with excitation energy. It is seen in Fig. 8,
that at a given excitation energy, the behavior of entropy is determined by
the increased complexity of the deformed configurations. Entropy is generally
higher at the SP than at the GS because the SP represents a more disordered
and diverse configuration of nucleons and benefits from using multiple energy
shells. The calculations at $U^{*}$ values corresponding to the considered
energies are shown with dotted lines. As seen, the structure effect is
stronger in this case. The damping at the GS is larger than that at larger
deformations. Hence, the enhancement of the entropy ratios at the SP seen from
the figure is both due to larger state density and slower potential damping.

Figure 8: (a) The ratios of entropies calculated at each deformation at fixed
excitation energies $U=15$ and 35 MeV to those of the GS with considering
potentials without damping (solid lines) for axial fission path. The
calculation at excitation energies with respect to the damped potentials
$U^{*}$ are presented with dotted lines. (b) the same as panel (a), but for
the triaxial path.

###  III.4 Fission probability

To investigate the impact of entropy on decay properties, we employ a
simplified approach to calculate the decay rates for two distinct paths. Our
analysis involves discretizing the trajectories into a finite number of grid
points: $i=(1:n_{\text{max}})$ for the axial path and $i=(-n_{\text{max}}:-1)$
for the triaxial path. The grid point $i=0$ corresponds to the configuration
of the GS. Each point is characterized by a set of deformation parameters
$\beta_{i}=(\beta^{(i)}_{20},\beta^{(i)}_{22})$

The probability $n(\beta_{i})$ of the nuclear system being populated at time
$t$ in a state characterized by deformation $\beta_{i}$ is determined by a set
of master equations Moretto1975

$\frac{dn(\beta_{i})}{dt}=\Lambda_{i,i+1}n(\beta_{i+1})+\Lambda_{i,i-1}n(\beta_{i-1})-\Lambda_{i\pm
1,i}n(\beta_{i}).$ (24)

We assume that only neighboring systems are connected. Consequently, within
our simplified model, systems belonging to different trajectories (axial or
triaxial) are only linked through the GS. This approximation seems appropriate
at low excitation energies. In Eq. (24), the symbol $\Lambda_{if}$ represents
the transition rates, which can be expressed in terms of microscopic
transition probabilities and the level density of the final states. Assuming
the same transition strength $\lambda_{0}$ in both axial and triaxial paths,
we calculate the transition width as

$\displaystyle\Lambda_{if}=\lambda_{if}\rho(\beta_{f}),$
$\displaystyle\lambda_{if}=\lambda_{fi}=\frac{\lambda_{0}}{\sqrt{\rho(\beta_{i})\rho(\beta_{f})}}.$
(25)

Here, we make an assumption that once the system reaches any of the SPs, it
certainly undergoes fission.

Since the system inevitably undergoes fission, the fission probabilities
through the SP of axial [$n^{SP}_{ax}(t)$] and triaxial [$n^{SP}_{tri}(t)$]
paths are connected as $n^{SP}_{ax}(\infty)+n^{SP}_{tri}(\infty)=1$. The
ratios of fission probabilities, corresponding to an axial and triaxial paths
calculated with and without accounting for damping of potential, are displayed
in Fig. 9 (a) as a function of $U_{0}$. The ratio of entropy at the SP of
axial fission path to that of triaxial fission path $S^{SP}_{ax}/S^{SP}_{tri}$
is also displayed in the panel (b). The contribution of the considered paths
to total fission depends on their relative entropies at the SP.

Figure 9: Panel (a): The ratio of fission probability through the SP of axial
fission path to that of triaxial fission path as a function of excitation
energy. Panel (b): The ratio of entropy at the SP of axial fission path to
that of triaxial fission path. The solid blue lines and dashed red lines
indicate the calculations with and without damping of potential, respectively.

The triaxial path exhibits dominance at lower energies, while at higher
energies, the contributions of both the axial and triaxial paths become
comparable. As seen from Fig. 9 (b), the $S^{SP}_{ax}/S^{SP}_{tri}$ ratios
both for the calculations with and without damping of potential are
approaching asymptotic values after $U_{0}=25$ MeV. Consequently, the ratios
of fission probabilities are expected to exhibit a constant behavior at
$U_{0}>25$ MeV. However, it is worth noting that when the damping effect is
taken into account, the entropy values from the GS to the SP get closer to
each other with increasing excitation energy so that, as shown in Fig. 7 (b),
at $U>25$ MeV they become almost constant. Hence, the assumption of two
distinct paths for the evaluation of fission probability becomes inadequate.
This finding suggests that in order to accurately model the fission process,
particularly when considering the damping effect, the competition between the
axial and triaxial paths should be accounted at all intermediate points along
the paths, rather than solely at the GS.

In the absence of damping, the GS entropy remains higher than that at the SP
even at higher excitation energies. This implies that there is a probability
for the system to return to the GS and subsequently have the opportunity to
choose another path. However, when the damping effect is taken into account,
because of relatively small difference between SP and GS entropies once the
system selects a path, it progresses toward the SP and does not revert back to
the GS. With damping of potential and decreasing of the variation of entropy
along the fission paths, the axial path, which has a SP closer to the GS,
achieves a non-zero decay probability more rapidly than the triaxial path.
This effect is represented in decay constants $\lambda$ which are determined
by fitting the $n^{SP}(t)$ with the following form

$n^{SP}(t)=n^{SP}(\infty)(1-e^{-\lambda t}).$

The ratio $\lambda_{tri}/\lambda_{tri}$ of decay constants corresponding to
the triaxial and axial pathways is displayed in Fig. 10. As shown, with
account of the damping effect the axial path reaches its asymptotic
probability $n^{SP}_{ax}(\infty)$ faster than triaxial path and this effect
increases with excitation energy.

Figure 10: The ratio of the decay constant corresponding to the triaxial path
to that of the axial path is shown as a function of excitation energy. The
solid and dashed lines indicate the calculations with and without damping of
potential, respectively.

##  IV CONCLUSIONS

Our analysis emphasizes that in the study of fission, it is essential to
consider not only the shape of the PES but also the entropy, which
incorporates both energy and structural effects. As demonstrated for the
superheavy nucleus 296Lv, variations in structural effects at different points
of the fission path can lead to significant changes of the level density and,
correspondingly, of the entropy, especially in the case of axial symmetry.
Furthermore, our work highlights the significant influence of suppressing of
shell effects in certain regions, where it is strong and cannot be neglected.
At sufficiently high excitation energies, the fission process exhibits iso-
entropic behavior. While, at lower energies, the entropy demonstrates more
pronounced variations. By employing the master equation, we computed the decay
probabilities that vary with time, considering the permissible symmetry in the
fission pathway. As shown, the time-dependent fission probability is
influenced by the behavior of entropy along the fission paths. The decay
constants corresponding to the axial and triaxial paths are comparable. To
provide a more precise analysis, fission calculations should be conducted on a
multi-dimensional PES, considering different trajectories leading to the SP
region. However, to gain a comprehensive understanding, one should expand the
study to encompass additional degrees of freedom and allowing the system to
select an optimal path. This extension will provide a more thorough
exploration of the fission process and its dependence on entropy.

## ACKNOWLEDGEMENTS

M. K. was co-financed by the COPIGAL project. T.M.S, G.G.A., and N.V.A. were
supported by Ministry of Science and Higher Education of the Russian
Federation (Moscow, Contract No. 075-10-2020-117).

## References

  * (1) Yu.Ts. Oganessian, J. Phys. G 34, R165 (2007).
  * (2) Yu. Ts. Oganessian and V. K. Utyonkov, Nucl. Phys. A 944, 62 (2015); Rep. Prog. Phys. 78, 036301 (2015).
  * (3) Yu. Ts. Oganessian and K. P. Rykaczewski, Physics Today 68, 32 (2015).
  * (4) S. Hilaire, J.P. Delaroche, and M. Girod, Eur. Phys. J. A 12, 169 (2001).
  * (5) S. Goriely, S. Hilaire, and A. J. Koning, Phys. Rev. C 78, 064307 (2008).
  * (6) A.H. Blin et al., Nucl. Phys. A 456, 109 (1986).
  * (7) H. Uhrenholt, S. Åberg, A. Dobrowolski, Th. Døssing, T. Ichikawa, and P. Möller, Nucl. Phys. A 913, 127 (2013); D.E. Ward, B.G. Carlsson, Th. Døssing, P. Möller, J. Randrup, and S. Åberg, Phys. Rev. C 95, 024618 (2017); M. Albertsson, B.G. Carlsson, Th. Døssing, P. Möller, J. Randrup, and S. Åberg, Phys. Lett. B 803, 135276 (2020).
  * (8) B. Nerlo-Pomorska, K. Pomorski, J. Bartel, and K. Dietrich, Phys. Rev. C 66, 051302(R) (2002).
  * (9) J. Bartel, K. Pomorski, and B. Nerlo-Pomorska, Int. J. Mod. Phys. E 15, 478 (2006); B. Nerlo-Pomorska, K. Pomorski, J. Bartel, and A. Dobrowolski, Acta Physica Polonica B 39, 417 (2008).
  * (10) R. A. Senkov and M. Horoi, Phys. Rev. C 82, 024304 (2010).
  * (11) Y. Alhassid, M. Bonett-Matiz, S. Liu, and H. Nakada, Phys. Rev. C 92, 024307 (2015).
  * (12) Yu. V. Sokolov, Level densities of atomic nuclei (Energoizdat, Moscow, 1990).
  * (13) A. B. Ignatyuk, G. N. Smirenkin, and A. S. Tishin, Yad. Fiz. 21, 485 (1975).
  * (14) A. Rahmatinejad, A. N. Bezbakh, T. M. Shneidman, G. Adamian, N. V. Antonenko, P. Jachimowicz, and M. Kowal, Phys. Rev. C 103, 034309, (2021).
  * (15) V. M. Strutinski, Sov. J. Nucl. Phys. 3, 449 (1966), Nucl. Phys. A 95, 420 (1967).
  * (16) S. Ćwiok, J. Dudek, W. Nazarewicz, J. Skalski, and T. Werner, Comput. Phys. Commun. 46, 379 (1987).
  * (17) H. J. Krappe, J. R. Nix, and A. J. Sierk, Phys. Rev. C 20, 992 (1979).
  * (18) I. Muntian, Z. Patyk, and A. Sobiczewski, Acta Phys. Pol. B 32, 691 (2001).
  * (19) P. Jachimowicz, M. Kowal, and J. Skalski, Phys. Rev. C 95, 014303 (2017).
  * (20) P. Jachimowicz, M. Kowal, and J. Skalski, Phys. Rev. C 101, 014311 (2020).
  * (21) P. Jachimowicz, M. Kowal, and J. Skalski, Atomic Data and Nuclear Data Tables 138, 101393, (2021). https://doi.org/10.1016/j.adt.2020.101393.
  * (22) P. Decowski, W. Grochulski, A. Marcinkowski, K. Siwek, and Z. Wilhelmi, Nucl. Phys. A 110, 129 (1968).
  * (23) G.D. Adeev and P.A. Cherdantsev, Yad. Fiz. 21, 491 (1975).
  * (24) F. A. Ivanyuk, C. Ishizuka, M. D. Usang, S. Chiba, Phys. Rev. C 97, 054331 (2018).
  * (25) M. Bolsterli, E. O. Fiset, J. R. Nix, and J. L. Norton, Phys. Rev. C 5, 1050 (1972).
  * (26) A. Rahmatinejad, A. N. Bezbakh, T. M. Shneidman, G. Adamian, N. V. Antonenko, P. Jachimowicz, and M. Kowal, Phys. Rev. C 105, 044328 (2022).
  * (27) L.G. Moretto and J.S. Sventek, Phys. Lett. B 58, 26 (1975).