Description of synthesis of super-heavy elements within the multidimensional stochastic model

We describe the fusion-fission processes within the two-stage reaction model. On the first stage (the entrance channel) the time evolution of the system up to the touching point is calculated. All possible orientations of the colliding nuclei are taken into account. The distributions at the touching point obtained on the first step are used for the transition from separated ions to the compact system and as the initial conditions for the description of evolution of compact system (second stage). Both the approach of ions and the evolution of the compact system are described with the help of Langevin equations for the collective coordinates (deformation parameters) which specify the configuration of the system. In both stages the shell structures of the colliding ions and the compound nucleus are taken into account. Within this model we obtain the information on the touching probability and the observable measured in the fusion-fission reactions (mass and energy distributions of the fission fragments, the compound nucleus, and the evaporation residue formation cross sections). In the present work we have carried out the calculations for the reactions ${}^{64}\mathrm{Ni}+{}^{238}\mathrm{U}\to {}^{302-x}120+xn$, ${}^{58}\mathrm{Fe}+{}^{244}\mathrm{Pu}\to$ ${}^{302-x}120+xn$, ${}^{54}\mathrm{Cr}+{}^{248}\mathrm{Cm}\to {}^{302-x}120+xn$, leading to the synthesis of an element with $Z=120$. The calculated results are compared with the available experimental data and other theoretical predictions.

Figure 1

Schematic representation of the system at the first stage of calculations. Parameters ${\alpha }_t$ and ${\alpha }_p$ are the deformation parameters of target and projectile nuclei, $r$ is the distance between their centers of mass, and ${\theta }_t$ is the angle between the symmetry axes of the target nucleus and the line connecting centers of mass of the colliding nuclei.

Figure 2

Dependence of the capture cross section on the center-of-mass energy $E_{\mathrm{c}.\mathrm{m}.}$ (top) and on the excitation energy $U^{*}$ (bottom) for a few reactions. Calculated results are shown by the lines. Open circles are the experimental data for the reaction ${}^{48}\mathrm{Ca}+$ ${}^{248}\mathrm{Cm}\to$ ${}^{296-x}116+xn$ [4].

Figure 3

Position of the initial deformation of the composite system with $Z=120$ on the potential energy surface $V_{\mathrm{pot}}({\alpha }_4,{\alpha }_1,\alpha =1)$ (in MeV) at the touching point.

Figure 4

Dependence of the potential energy $V_{\mathrm{pot}}(\alpha ,{\alpha }_1,{\alpha }_4)$ (in MeV) of the composite system with $A=302$, $Z=120$ on the deformation parameters $\alpha ,{\alpha }_4$ at ${\alpha }_1=0$. Solid line shows the boarder of stability with respect to the neck rupture; dashed line is the position of the fission barrier; dotted line is the trajectory towards the ground state deformation.

Figure 5

Yield of the quasi-fission fragments $Y$ (normalized to 200%) in the reaction ${}^{48}\mathrm{Ca}+{}^{248}\mathrm{Cm}\to {}^{296}{\mathrm{Xx}}_{116}$ at excitation energy $U^{*}=37$ MeV. The experimental data (open points) are taken from [23]. The calculated values are shown by the solid line.

Figure 6

Cross sections of crossing fission barrier (top) and evaporation residue formation (bottom) for $Z=116$ (solid) and $Z=120$ (dash, dash-dot, dot). Open circles mark the experimental results for the reaction ${}^{48}\mathrm{Ca}+{}^{248}\mathrm{Cm}\to$ ${}^{296-x}116+xn$. Dot and dash-dot arrows show the limiting values for ${}^{64}\mathrm{Ni}+{}^{238}\mathrm{U}\to {}^{302-x}120+xn$ ($U^{*}=36.4\mathrm{MeV}$) and ${}^{58}\mathrm{Fe}+{}^{244}\mathrm{Pu}\to {}^{302-x}120+xn$ ($U^{*}=45.5\mathrm{MeV}$) reactions, respectively.