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\title{Asymmetric Dynamical Simulation of Energy Dissipation in Heavy-Ion Induced Fission of ${^{200}Pb}$, ${^{213}Fr}$ and ${^{251}Es}$}

\author{M.\ R.\ Pahlavani% } \author{S.\ M.\ Mirfathi% \footnote{Email: m.mirfathi@umz.ac.ir} } \affiliation{ $^{\mathrm{a}}$Department of Physics, Faculty of Basic science, Mazandaran University, P.O.Box 47416-1467, Babolsar, Iran\\ } %\received{xx xxxxxxx xxxx} %\accepted{xx xxxxxxx xxxx}

\begin{abstract} The asymmetric dynamical model is applied to calculate pre-scission neutron multiplicity from heavy-ion induced fusion-fission reactions. We discuss the link between the pre-scission neutron multiplicity, excitation energy and asymmetric mass distribution. Our approach is based on the Monte Carlo simulation and Langevin dynamics. The pre-scission neutron multiplicities have been calculated and compared with the respective experimental data over a wide range of excitation energy. The analysis provides a different effects for the application of asymmetric mass division in systems with different fissility parameters. \end{abstract}

\pacs{25.70.Jj,05.10.Gg,21.60.Ka} \keywords{Fusion-Fission;Langevin Equations;Monte Carlo}

\maketitle \section{Motivation} In the recent century the development of nuclear accelerators has been prepared Heavy-Ion Beams for fusion-fission reactions\cite{l00005}, and opened an opportunity in study of fission process induced by heavy ion. Besides the distance between mass centers of nascent fragment, the asymmetry parameter and and shape elongation play a crucial role in study of fusion- fission and deep inelastic processes of specially low energy heavy ion collisions\cite{l2}. We studied the whole dynamics of such a nuclear system by using Monte Carlo approach to solve the coupled Langevin-type equations of motion. A number of early studies of pre-scission neutron multiplicity and energy dissipation of such reactions can be found in the literature\cite{l00005,l2,l3,l1000,l100,l10,l1,l0005}. For instance, Frobrich and Gontchar\cite{l005}, in an approach based on dissipation effects, used Langevin equations to to estimate the energy dissipated in heavy ion fusion-fission reactions and number of neutrons that emitted during descent from saddle to scission. Later, Chaudhury and Pal discussed\cite{l05}, for an average light and heavy fragment, the competition of neutron and $\gamma-ray$ emission in fission based on dynamical approach taking into account some different. Recently, Lemaire and co-workers developed a theory to study the prompt fission neutron and $\gamma-ray$ evaporation process, where they are emitted sequentially from fragments \cite{l006,l06}. However our Monte Carlo approach allows us to compare our results with various prompt fission neutrons observables such as energy of emitted neutrons, neutron multiplicity distribution and so on. The study of neutron emission accompanying asymmetric fission has highlighted the important role of dissipation in both statistical and dynamical approaches. It should be noted that, in general for light nuclei with fissilities below Businaro-Gallone point an asymmetric mass splitting is favored\cite{l5,l6}. We first present the theoretical methodology and give the input parameters that enter into our calculation. This is followed by a presentation and discussion of the results for three systems with different fissilities. Our model is applied to the reaction systems ${^{30}Si}+{^{170}Er}$ , ${^{16}O}+{^{197}Au}$ and ${^{19}F}+{^{232}Th}$ with a wide range of the incident energy corresponding to the excitation energy of the compound nucleus. A summary and conclusion of future improvements of this work completes this paper.

\section{THEORETICAL APPROACH} We will recall in the next sections some of the main steps of our approach. A Monte Carlo technique is used to simulate compound nucleus deexcitation. Our strategy differs with existing literatures in one main respect: we consider possible asymmetry for all nascent fragments and then we calculate pre-scission neutron multiplicity for a broad range of asymmetric fragments, finally average of these multiplicities show pre-scission neutron multiplicity for that system. Also we used a modified combined Langevin dynamics plus a general dissipation relation for our systems. In the following we discuss the method by the specific examples on different systems of heavy-ion reactions.

\subsection{Asymmetric Dynamical Model [ADM]} The dynamical time evolution of the fission process and energy deexcitation of compound nucleus during descent from saddle to scission constitute a complex problem. Considering "funny hills" parametrization $(c,h,\alpha)$ \cite{l7} and its conjugate momentum $p$ as the dynamical variables, the Langevin equations in three dimension will be given\cite{l05} \begin{eqnarray} & & \hspace{-7.7cm}\frac{dq_i}{dt}=\frac{p_j}{m_{ij}}\\ \hspace{-0.9cm}\frac{dp_i}{dt}=-\frac{p_i\,p_j}{2}\,\frac{\partial}{\partial q_i}\,(\frac{1}{m_{ij}})-\frac{\partial F}{\partial q_i}-\gamma_{i}\,\frac{dq_i}{dt}+R(t) \end{eqnarray} In the above equations $q=(c,h,\alpha)$ are the collective coordinates, $p=(p_c,p_h,p_{\alpha})$ are the conjugate momenta. $m$ and $\gamma$ are the shape-dependent collective inertia and friction coefficients respectively. Here $i,j$ are refers to funny hills parameters. The neck thickness and asymmetry parameter denoted by $h$ and $\alpha$ as follows\cite{l188} \begin {equation} h=-1.047\,c^3+4.297\,c^2-6.309\,c+4.073 \end {equation} and \begin {equation} \alpha=0.11937\alpha_{asy}^2+0.24720\alpha_{asy} \end {equation} which $\alpha_{asy}=(A_1-A_2)/A_{C.N.}$. Also $c$, $F$ and $R(t)$ represents the elongation, free energy of the system and random part of the interactions between the fission degree of freedom and thermal bath\cite{l9}. Following the work of Frobrich and Gontchar\cite{l005}, we solved these equations by coupling they with neutron and $\gamma$ emission at each time step evolution of fission plus a Monte Carlo scheme that are often the only practical way to evaluate such difficult integrals. For parametrization of the nuclear surface we used a parametric family of shapes based on cylindrical coordinates as follows \begin{equation} \rho_s^2(z)=(c^2-z^2)(A_s/c^2+B_{sh}\,z^2/c^2+\alpha \,z) \end{equation} where $\rho_s$ is the radial coordinate of the nuclear surface and $z$ is the coordinate along the symmetry axis. $A_s$ and $B_{sh}$ are defined in Ref.\cite{l188}. It is already pointed out in the literature that one-body dissipation is the dominant mode of energy damping in nuclear fission in describing fission dynamics\cite{l11}, however for a general description we used both one and two-body dissipation. For do this we used following expressions to calculate the two and one-body frictions \cite{l188,l020,l0222} \nonstopmode \begin{equation} \gamma_{i}^{TB}= \pi\,\mu_0\,R_{C.N.}\,f_i\,\int_{-c}^{+c}\rho_s^2(z)[3{A'}_i^2+\rho_s^2(z)\,{A''}_i^2/8]\,dz \end{equation} and \begin{eqnarray} \gamma_{i}^{OB}&=& 2\pi\,\rho_m\,\bar{v}\,R_{C.N.}^2\,f_i \nonumber\\ & & \hspace{-0.9cm}\times\int_{-c}^{+c}\rho_{s}(z)[A_i\,\rho'_s+A'_i\,\rho_s/2]^2\,[1+{\rho'}_s^2]^{-1/2}\,dz \end{eqnarray} Where $\mu_0$ is the viscosity coefficient, $R_{C.N.}$ is the radius of compound nucleus and $f_i$ defined as follows\cite{l188} \begin {equation} f_i=(\frac{\partial\,q_i}{\partial\,x})^2+2\,\frac{\partial\,q_i}{\partial\,x} \end {equation} Here $x=r_{cm}/R_{C.N.}$ and the parameter $r_{cm}$ is defined as the center to center distance between the two part of fissioning system. Also $A_i(z)$ defined as follows \begin {equation} A_i(z)=-\frac{1}{\rho^2{z}}\,\frac{\partial}{\partial q_i}\,\int_{-c}^{z}\rho^2{z'}dz' \end {equation} The quantities $A'_i$ and $A''_i$ are the first and second derivatives of $A_i(z)$ with respect to $z$\cite{l188}. $\rho_m$is the nuclear density, $\bar{v}$ is the average nucleon speed inside the nucleus\cite{l188} and $\rho_{s}^{'}$ is the first derivative of $\rho_{s}$ with respect to $z$ also $i$ is refer to funny hills parameters. It is natural that for measuring the competition between neutron,$\gamma-ray$ and fission we need their decay width. Neutron decay width at that instant is calculated as follows\cite{l13} \begin{equation} \Gamma_{n}=\frac{2 m_n}{(\pi\hbar)^2 \rho_m(E_{int})}\int_{0}^{E_{int}-B_n}d\varepsilon \rho_d(E_{int}-B_n-\varepsilon) \varepsilon \sigma_{inv} \end{equation} Where $\varepsilon$ is the energy of emitted neutron and $\rho_m$ and $\rho_d$ are the level densities of the compound and residual nuclei that are denoted by\cite{l14} \begin{equation} \rho(E_{int},A,l)=\frac{(2 \hbar l+1)\sqrt{a}}{12 E_{int}^2} [\frac{\hbar^2}{2 J_0}]^{3/2} exp(2 \sqrt{a E_{int}}) \end{equation} Where $\hbar l$ is the angular momentum of the compound or residual nuclei, $\sigma_{inv}$ is the inverse cross section\cite{l15}, $B_n$ is the binding energy of neutron and the level density parameter denoted by $a$. Also $E_{int}$ is the intrinsic excitation energy of the parent nucleus and $J_0$ is the moment of inertia \cite{l005}. The $\gamma-quanta$ decay width at each time step is calculated using the following relation\cite{l005} \begin{equation} \Gamma_{\gamma}=\frac{3}{\rho_m(E^*)}\int_{0}^{E_{int}}d\varepsilon \rho_d(E_{int}-\varepsilon) f(\varepsilon) \end{equation} Where $\varepsilon$ is the energy of emitted $\gamma-ray$ and $f(\varepsilon)$ denoted by\cite{l005} \begin{equation} f(\varepsilon)=\frac{0.74 e^2 N Z}{A \hbar m c^3} \frac{5 \varepsilon^4}{(5 \varepsilon)^2+(\varepsilon^2-(80 A^{-1/3})^2)^2} \end{equation} Here $A$, $Z$ and $N$ refers to compound nucleus.

\subsection{Monte Carlo Dynamical Simulation} In the reaction process, we first specify the entrance channel through which a compound nucleus is formed and assuming complete fusion of the target with the projectile. The initial spin of the compound nucleus will be obtained by sampling the fusion spin distribution\cite{l005}. As a fully equilibrated compound nucleus is formed at a certain instant that is fixed as the origin of our dynamical trajectory calculation and following earlier literature we assumed that the initial distribution of the coordinates and momenta are chosen from sampling random numbers following the Maxwell-Boltzmann distribution\cite{l005,l9,l188}. The process of neutron and $\gamma$ emission from a compound nucleus is governed by the emission rate such as equations (9)and (11). It should be noted that several approaches have been used in earlier works to describe the emission of particles such as neutrons from a highly excited nucleus, that we used here Weisskopf’s conventional evaporation theory for neutrons\cite{l13} and Lynn theory for the emission of giant dipole $\gamma$-quanta\cite{l17}. The Monte Carlo algorithm used to calculate competition between neutron emission,$\gamma$-quanta emission and fission. The widths of neutron,$\gamma$ and fission decay rate depend upon the excitation energy, spin and the mass number of the compound nucleus and hence are to be evaluated at each interval of time evolution of the fissioning nucleus due to Langevin equations. To do this we first choose a random number $r$ on the half open interval $[0,1)$ by using Monte Carlo techniques. The random number is a numerical characteristic assigned to an element of the sample space. Then we define the probability of emission of a neutron as $x=\tau/\tau_n$,where $\tau_n$ is the neutron decay time and $\tau$ is the time step of the calculation. If $r<x$, it will be interpreted as a particle emission. Following the same procedure the type of the emitted particle is decided by a Monte Carlo selection based on the law of radioactive decay for the emitted particles. After each emission the intrinsic excitation energy of residual compound nucleus recalculated due to the energy that released based on the one particle emission. Also other parameters such as mass and spin of the compound nucleus are recalculated after each emission. This circle of calculation repeated until $c=c_{sci}$ \begin{equation} c_{sci}=-2.0\,\alpha^2+0.032\,\alpha+2.0917 \end{equation} As stated earlier we calculate pre-scission neutron multiplicity for each $\alpha$, finally average of these multiplicities show pre-scission neutron multiplicity for that system. Therefore average pre-scission neutron multiplicity is given by \begin{equation} =\frac{\sum_{\alpha}\,\sum_{l}\,_{l,\alpha}\,(2l+1)\,P_l}{\sum_{\alpha}\sum_{l}\,(2l+1)P_l} \end{equation} Where the probability to cross the fission barrier which depend upon angular momentum denoted by $p_l$. \begin{equation} p_l=\frac{N_l}{N} \end{equation} Here $N$ and $N_l$ are the total number of trajectories and the number of trajectories which undergo fission respectively. Summation of $\alpha$ defined in an interval $[0,\alpha_f]$ and summation of $l$ defined in an interval $[0,l_f]$, Which $\alpha_f$ and $l_f$ refers to maximum asymmetry and critical angular momentum for fusion respectively.

\section{Results}

We have mainly considered heavier mass nuclei since it is for these nuclei that the neutron and fission widths become comparable and their competition strongly dictates the final observables. By considering the alternative Monte Carlo, and directly comparing the corresponding neutron multiplicity of various systems, we study the influence of asymmetry parameters on deexcitation of energy. To illustrate these concepts, we have studied ${^{30}Si}$ induced fission of ${^{170}Er}$ and ${^{200}Pb}$ as compound nucleus, then ${^{213}Fr}$ that formed after complete fission of ${^{16}O}$ as projectile with ${^{197}Au}$ as target, and finally ${^{19}F}$ induced fission of ${^{232}Th}$ and ${^{251}Es}$ as compound nucleus with fissility parameters $33.62$,$35.54$and $39.05 $ respectively. \begin{figure} \centering \mbox{\includegraphics[width=8.0cm]{fig1.eps}} \caption{ Variation of pre-scission neutron multiplicity versus excitation energy in our calculation(dashed line) compare with earlier experimental and theoretical results\cite{l005} that are shown with filled squares and triangles for ${^{200}Pb}$. } \end{figure} \begin{figure} \centering \mbox{\includegraphics[width=8.0cm]{fig2.eps}} \caption{ Pre-scission neutron multiplicity of ${^{213}Fr}$ as function of excitation energy. Results of our calculation(dashed line)compare with earlier experimental and theoretical results(filled squares and triangles)\cite{l005}. } \end{figure} \begin{figure} \centering \mbox{\includegraphics[width=8.0cm]{fig3.eps}} \caption{ Variation of pre-scission neutron multiplicity due to excitation energy. Results of our calculation(dashed line) compare with earlier experimental and theoretical results (filled squares and triangles)\cite{l005} for ${^{251}Es}$.} \end{figure}

\section{Summary and Conclusion} We have developed a combined statistical and dynamical model for fission where fission trajectories are generated by solving Langevin equations of motion using a combination of one and two-body dissipations. The choice of a Monte Carlo simulation to describe this processes allows us to infer important physical quantities that could not be assessed otherwise, and can also be used to assess in particular the validity of assumptions about how does the available total excitation energy get distributed among the asymmetric fragments\cite{l3}. In summary, we proposed a model based on both asymmetric statistical and dynamical approaches to reproduce neutron multiplicity. Among the various physical inputs required for solving the Langevin equation with Monte Carlo algorithm, we paid more attention to the mass distribution of fragments. As shown in figures 1 to 3 in medium energy and for heavier systems with larger fissility we see agreements between our calculation with earlier results. To conclude, in the present model and in medium excitation energy with the generalized shape parametrization and combined dissipation, both symmetric and asymmetric splitting of the compound nucleus can be treated on the same footing. Also despite the strong friction on the collective motions of nucleons before scission both earlier statistical calculation and our approach gives good equality with experimental neutron multiplicity specially at medium energies and large fissility. We gave a general overview on the mass distribution in nuclear induced fission brought about by the new data, that contain much more information relevant for the understanding of this process than could be touched in this paper. The pre-scission neutron multiplicity is found to be higher than that calculated using a statistical saddle point model, which is consistent with the observations at near barrier energies for other systems. Finally the theoretically pre-scission neutron multiplicity , averaged over the wide range of mass distribution, was found to be significantly higher than that calculated using a statistical saddle point model. With all the experimental progress in the heavy-ion fusion-fission reactions, what can we anticipate for the future ? Of course there will be a steady improvement in the precision and confidence with which we can determine the appropriate fission model and its parameters. \section{Acknowledgements} We would like to thank Dr. J. Sadeghi for numerous enlightening discussions and critical comments that largely determined the contents of the present paper. \begin{thebibliography}{99} \bibitem{l00005} H. Rossner {\it et al.} Phys. Rev. C 45, 719 (1992). \bibitem{l2} V. I. Zagrebaev,Y.T. Oganessian, M.G.Itkis, W. Greiner Phys. Rev. C 73, 031602(R)(2006). \bibitem{l1000} P. Fong Phys. Rev. 102, 434 (1956). \bibitem{l100} P. N. Nadtochy and G. D. Adeev Phys. Rev. C 72, 054608 (2005). \bibitem{l10} M. Schadel {\it et al.} Phys. Rev. Lett. 48, 852 (1982). \bibitem{l1} K. J. Moody {\it et al.} Phys. Rev. C 33, 1315 (1986). \bibitem{l3} S.Lemaire, P.Talou, T. Kawano, M.B. Chadwick, D.G. Madland Phys. Rev. C 72, 024601 (2005). \bibitem{l0005} D. J. Hinde {\it et al.} Phys. Rev. C 45,1229(1992). \bibitem{l005} P. Frobrich and I. I. Gontchar Phys. Rep. 292 ,131(1998). \bibitem{l05} G. Chaudhuri,and S. Pal Phys. Rev. C 65, 054612(2002). \bibitem{l006} S. Lemaire, S.Lemaire, P.Talou, T. Kawano, M.B. Chadwick, D.G. Phys. Rev. C 73, 014602(2006). \bibitem{l06} S. Lemaire {\it et al.} Phys. Rev. C 72, 024601(2005). \bibitem{l5} C. Beck {\it et al.} Z.Phys.A 334,521(1989). \bibitem{l6} C. Beck {\it et al.} Phys. Rev. C 47, 2093 (1993). \bibitem{l7} G.D.Adeev and P.A.Cherdantsev Phys. Lett B 39, 485 (1972). \bibitem{l188} A.K.Dhara,K. Krishan, C.Bhattacharya, S. Bhattacharya Phys. Rev. C 57, 2453(1998). \bibitem{l9} G. Chaudhuri,and S. Pal Phys. Rev. C 63, 064603(2001). \bibitem{l11} T.Wada,Y. Abe, N. Carjan Phys. Rev. Lett. 70, 3538(1993). \bibitem{l020} K.T.R.Davis {\it et al.} Phys. Rev. C 13, 2385(1976). \bibitem{l0222} H.Feldmeier {\it et al.} Rep. Prog. Phys. 50, 915(1987) \bibitem{l13} V. Weisskopf Phys. Rev. 52, 295 (1937). \bibitem{l14} A. Bohr and B.R. Mottelson, Nuclear structure, Vol. II, Benjamin, London, 1975. \bibitem{l15} M. Blann, Beckerman Nucleonika 23, 34(1978). \bibitem{l17} J.E. Lynn, Thoery of Neutron Resonance Reactions, Clarendon, Oxford, (1968) p. 325.

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