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\title{\bf Three fission modes in ${}^{258}$Md studied by Langevin model}
%%%%%%%%%%%%%%%%%%%%%authors%%%%%%%%%%%%%%%%%%%%%
\author[1]{Ishizaki Shoma$^\dagger$}
\author[1]{Yoshihiro Aritomo}
\author[1]{Mizuki Okubayashi}
\author[2]{Katsuhisa Nishio}
\author[2]{Kentarou Hirose}
\affil[1]{Faculty of Science and Engineering, Kindai University Higashi-Osaka,
Osaka 577-8502, Japan}
\affil[2]{Advanced Science Research Center,
Japan Atomic Energy Agency Tokai, Ibaraki 319-1195, Japan}
\affil[ ]{$^\dagger$Email: uthorium232@gmail.com}
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\date{}
\maketitle
%%%%%%%%%%%%%%%%%%%%%abstract%%%%%%%%%%%%%%%%%%%%%
\begin{abstract}
Recently, fission-fragment mass properties of ${}^{258}$Md was measured in JAEA. The data indicates a mixture of fission modes.
We tried to calculate the fission properties of ${}^{258}$Md using the Langevin model.
\end{abstract}
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%%%%%%%%%%%% main %%%%%%%%%%%%
\section{Introduction}
\par It has been shown that fission has multiple modes, characterized by mass asymmetric fission and mass symmetric fission [1-7]. In neutron-rich heavy element region, it is argued that several fission modes coexist, with a significant change of their yields in accordance with the number of neutrons contained in the fissionig nucleus. A typical example is found in the fermium isotopes.
The dominant mode transition is from the asymmetric splitting for ${}^{257}$Fm to the shape symmetric one for ${}^{258}$Fm [8].
This transitions was interpreted as due to the lowering of the fission barrier for symmetric fission toward heavier mass isotopes.
It’s important to know the potential energy surface structure and nuclear's deformation process to understand fission mechanism in neutron-rich heavy element region [9].
\par At the JAEA tandem facility, fission of an exited compound nucleus ${}^{258}$Md was studied in the reaction of ${}^{4}$He+${}^{254}$Es. Based on the systematics of the spontaneous fission [8], the nucleus are located in the region where mass-symmetric fission dominates.
\par For the discussion of the experimental data, we made a calculation of fission using the Langevin model. For the later discussion, we define three types of fission paths (modes); mass-asymmetric fission (standard mode), symmetric fission with high total kinetic energy TKE (supershort), and symmetric fission with low TKE (superlong).
%%%%%%%%%%%%%%%%%%%%%Framework%%%%%%%%%%%%%%%%%%%%%
\section{Framwork}
\par We use the fluctuation-dissipation model and employ the Langevin equations[10] to investigate the fission process.
The nuclear shape is defined by the two-center parametrization [11,12], which has three deformation parameters,
$z_0$, $\delta$, and $\alpha$ to serve as collective coordinates: $z_0$ is the distance between two potential centers,
$\alpha$ is a mass-asymmetry parameter defined by ($A_{1}$-$A_{2}$)/($A_{1}$+$A_{2}$) using fragment masses, $A_{1}$ and $A_{2}$.
The symbol $\delta$ denotes the deformation of the fragments defined as $\delta$ = 3($R_{\parallel}$−$R_{\perp}$)/(2$R_{\parallel}$+$R_{\perp}$),
where $R_{\parallel}$ and $R_{\perp}$ are the half length of the axes of an ellipse in the $z_0$ and $\rho$ directions of the cylindrical coordinate,
respectively, as shown in Figure 1 in Ref. [10].
\par We adopted the neck parameter $\varepsilon$=0.55 following the empirical relation in Ref. [9].
The three collective coordinates are abbreviated as $q$, $q$ = {$z$,$\delta$,$\alpha$}.
For a given value of a temperature of a system $T$, the potential energy is defined as a sum of the liquid-drop (LD) part,
a rotational energy and a microscopic (SH) part:
\begin{flalign}
V(q,l,T)&=V_{LD}(q)+\frac{\hbar^2l(l+1)}{2I(q)}+V_{SH}(q,T), \\
V_{LD}(q)&=E_{s}(q)+E_{c}(q),\\
V_{SH}(q,T)&=E^0_{shell}(q)\Phi(T), \\
\Phi(T)&=\exp\left({-\frac{aT^2}{E_d}}\right).
\end{flalign}
Here, $V_{LD}$ is the potential energy calculated with the finite-range liquid drop model,
given as a sum of the surface energy $E_{S}$ [12] and the Coulomb energy $E_{C}$.
$V_{SH}$ is the shell correction energy evaluated by the Strutinski method from the single-particle levels of the two-center shell model.
The shell correction has a temperature dependence expressed by a factor $\Phi$(T), in which $E_{d}$ is the shell damping energy chosen to be 20 MeV [13] and $a$ is the level density parameter.
At the zero temperature $(T=0)$, the shell correction energy reduces to that of the two-center shell model values $E^{0}_{shell}$.
The second term on the right-hand side of Eq. (1) is the rotational energy for an angular momentum $l$ [10], with a moment of inertia at $q$, $I$($q$).
\par The multidimensional Langevin equations [10] are given as
\begin{flalign}
\frac{dq_{i}}{dt}&=\left(m^{-1}\right)_{ij}p_{i},\\
\frac{dp_{i}}{dt}&=-\frac{\partial V}{\partial q_i}-\frac{1}{2}\frac{\partial}{\partial q_i}\left(m^{-1}\right)_{jk}p_{j}p_{k}-\gamma_{ij}\left(m^{-1}\right)_{jk}p_{k}+g_{ij}R_{j}(t),
\end{flalign}
where $i$ =\{$z$,$\delta$,$\alpha$\} and $p_{i}$ = $m_{ij}dq_{j}/dt$ is a momentum
conjugate to coordinate $q_i$ . The summation is performed over
repeated indices. In the Langevin equation, $m_{ij}$ and $\gamma_{ij}$ are
the shape-dependent collective inertia and the friction tensors, respectively. The wall-and-window one-body dissipation [14-16] is adopted for the friction tensor which can describe
the pre-scission neutron multiplicities and total kinetic energy
of fragments simultaneously [17]. A hydrodynamical inertia
tensor is adopted with the Werner-Wheeler approximation for
the velocity field [18]. The normalized random force $R_{i}$(t)
is assumed to be that of white noise, i.e., ⟨$R_{i}(t)$⟩=0 and
⟨$R_{i}(t)R_{j}(t)$⟩=2$\delta_{ij}\delta(t_{1}-t_{2})$. The strength of the random force $g_{ij}$ is given by the Einstein relation
$\gamma_{ij}T=\sum_{k}g_{ij}g_{jk}$.
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%%%%%%%%%%%%%%%Results and discution%%%%%%%%%%%%%%%
\section{Results and discussion}
% \subsection{Calculation results FFMDs and TKE distribution}
Figure 1 shows the calculated results of the FFMDs and TKE distributions of ${}^{258}$Md. FFMDs show mass symmetric splitting.
The peak of the TKE distribution is about 235 MeV. From features, the supershort mode was dominant in the calculation results.
\par Figure 2 shows the evolution of each fission modes as a function of excitation energy of ${}^{258}$Md (in this work, fission modes was defined in the range of mass numbers and TKE in Table 1). Here, the count as the vertical axis means the number of events in the Langevin calculation. The calculation shows that the standard mode decreases with excitation energy. It is considered that this is because the shell structure responsible for mass-asymmetry decreases with excitation energy.
\par Figure 3 shows the fission pathways of each mode in the $z-\delta$ plane. From the 1st local minimum(around \{$z$,$\delta$\}$\sim$\{0.0,0.2\}) to the 2nd local minimum(around \{$z$,$\delta$\}$\sim$\{1.0,0.2\}), the three modes behave very similarly. But it can be seen that the standard mode branches first at $z\approx1.3$ and then the superlong mode branches at $z\approx1.8$.
\par Figure 4 shows the energy of the fissioning nucleus at each shape plotted as a function of z. In Figure 4, the saddle point is located at $z\approx1.3$ in the standard mode, $z\approx1.8$ in the superlong mode, and $z\approx2.0$ in the supershort mode, so it was found that the branch point shown in Figure 3 is located at the saddle point. In particular, the saddle point of superlong mode is close to that of supershort mode, the trajectory is very similar up to the saddle point of superlong mode.
\begin{figure}[h]
\begin{center}
\includegraphics[width=130truemm]{calonly18MeV.png}
\caption{Calculation result of FFMDs(left) and TKE distribution(right) at excitation energy $E^{*}=18MeV$}
\end{center}
\end{figure}
\begin{table}[htbp]
\caption{Mass and TKE region of each mode}
\centering
\begin{tabular}{clll}
\hline
mode & Mass & TKE \\
\hline \hline
supershort & 114$\leqq${Mass}$\leqq$144 & TKE$>$220MeV \\
superlong & 114$\leqq${Mass}$\leqq$144 & TKE$\leqq$220MeV \\
standard & 114$>$Mass,144$<$Mass & TKE$\leqq$220MeV \\
\end{tabular}
\end{table}
\begin{figure}[h]
\begin{center}
\includegraphics[width=72truemm]{mode_trans.png}
\caption{The counts of each mode as a function excitation energy $E^{*}$}
\end{center}
\end{figure}
\begin{figure}[h]
\begin{minipage}{0.5\hsize}
\begin{center}
\includegraphics[width=66truemm]{2ndlocalminimum.png}
\caption{Each mode trajectory in $z-\delta$ plane \\ on $\alpha=0$ potential energy surface \\
(The white square in the figure indicates the second minimum point) }
\end{center}
\end{minipage}
\begin{minipage}{0.5\hsize}
\begin{center}
\includegraphics[width=65truemm]{18MeVz-v.png}
\caption{Each mode trajectory in $z-V$ plane}
\end{center}
\end{minipage}
\end{figure}
\section{Conclusion}
In this work, the fission mode of ${}^{258}$Md was studied by trajectory calculation using a Langevin equations. As a result, it was found that
the investigated three modes bifurcate around the exit point of the 2nd local minimum on potential energy surface.
Also, since the saddle point of superlong
mode is close to that of supershort mode, the trajectory is very similar up to the saddle point
of superlong mode.
\section*{Acknowledgements}
The Langevin calculations were performed using the cluster computer system (Kindai-VOSTOK) which is supported
by Research funds for External Fund Introduction by Kindai University.
\begin{thebibliography}{99}
\setlength{\itemsep}{-2pt}
\bibitem{1}E. K. Hulet, J. F. Wild, R. J. Dougan, R. W. Lougheed, J. H. Landrum, A. D. Dougan,
\bibitem{2}P. A. Baisden, C. M. Henderson, R. J. Dupzyk, R. L. Hahn, M. Schädel, K. Sümmerer, and G. R. Bethune, Phys. Rev. C 40, 770 (1989).
\bibitem{3}V. Pokrovsky, L. Calabretta, M. G. Itkis, et al., Phys. Rev. C 60, 041304
\bibitem{4}E. K. Hulet, R. W. Lougheed, J. H. Landrum, J. F. Wild, D. C. Hoffman, J. Weber, and J. B. Wilhelmy, Phys. Rev. C 21, 966 (1980).
\bibitem{5}J.F. Wild, J. van Aarle, W. Westmeier, R. Lougheed, E.K. Hulet, K.J. Moody, R.J. Dougan, E.-A. Koop, R.E. Glaser, R. Brandt and P. Patzelt, Phys. Rev., C 41, 640 (1990).
\bibitem{6}S. Cwiok P. Rozmej, A. Sobiczewski, and Z. Patyk, Nucl. Phys. A 491, 281 (1989).
\bibitem{7} U. Brosa, S. Grossmann, and A. Muller, Phys. Rep. 197,167 (1990).
\bibitem{8} D.C. Hoffman et al., Phys. Rev. C, 21, 1980 (637)
\bibitem{9} Y. Miyamoto et al., Phys. Rev. C, 99, 051601(R) (2019).
\bibitem{10} Y. Aritomo and M. Ohta, Nucl. Phys. A 744, 3 (2004).
\bibitem{11} K. Sato, A. Iwamoto, K. Harada, S. Yamaji, and S. Yoshida,Z. Phys. A 288, 383 (1978).
\bibitem{12} H. J. Krappe, J. R. Nix, and A. J. Sierk, Phys. Rev. C 20, 992(1979).
\bibitem{13} A. V. Ignatyuk, G. N. Smirenkin, and A. S. Tishin, Sov. J. Nucl. Phys. 21, 255 (1975).
\bibitem{14} J. Blocki, Y. Boneh, J. R. Nix, J. Randrup, M. Robel, A. J. Sierk, and W. J. Swiatecki, Ann. Phys. 113, 330 (1978).
\bibitem{15} J. R. Nix and A. J. Sierk, Nucl. Phys. A 428, 161c (1984).
\bibitem{16} H. Feldmeier, Rep. Prog. Phys. 50, 915 (1987).
\bibitem{17} T.Wada,Y.Abe, and N. Carjen,Phys. Rev. Lett. 70, 3538 (1993).
\bibitem{18} K. T. R. Davies, A. J. Sierk, and J. R. Nix, Phys. Rev. C 13, 2385
(1976).
\end{thebibliography}
\end{document}