Study of the dinuclear system for 296119 superheavy compound nucleus in the fusion reactions

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J. Mohammadi and O. N. Ghodsi. Study of the dinuclear system for 296119 superheavy compound nucleus in the fusion reactions[J]. Chinese Physics C.
J. Mohammadi and O. N. Ghodsi. Study of the dinuclear system for 296119 superheavy compound nucleus in the fusion reactions[J]. Chinese Physics C. shu
Received: 2020-10-30
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Study of the dinuclear system for 296119 superheavy compound nucleus in the fusion reactions

    Corresponding author: J. Mohammadi,
  • Department of Physics, Faculty of Science, University of Mazandaran, P.O. Box 47415-416, Babolsar, Iran

Abstract: This investigation seeks to find an appropriate dinuclear system for the formation of ${}^{296}$119 superheavy compound nucleus. By studying driving potential and measuring the capture cross section of the reactions, the evolution of the dinuclear system can be understood. In this study, we have obtained capture, fusion and evaporation residue cross section and the survival probability at energies near the Coulomb barrier for four reactions consists of: $^{45}$Sc + $^{251}$Cf, $^{42}$Ca + $^{254}$Es, $^{39}$K + $^{257}$Fm, and $^{38}$Ar + $^{258}$Md. Our calculations show that the reaction $^{38}$Ar + $^{258}$Md is a suitable choice for the formation of an element with 119 protons among the studied reactions from the theoretical viewpoint.


    • We know that the half-life of elements decreases with increasing atomic number, so the synthesis of superheavy elements are challenging. On the other hand, nuclear research has foreseen the existence of a so-called “Stability Island” for certain superheavy elements of the nuclide chart, which ought to have half-lives starting from several minutes to several years. Achieving this stability island will produce super-heavy nuclei with a half-life long enough to do real experiments [1].

      Superheavy nuclei are produced through a fusion action in the heavy ion collisions with different methods. One of these methods involves the concept of the dinuclear system (DNS) model, which works well to describe the fusion in reactions producing superheavy nuclei [2-7]. At energies around the Coulomb barrier for heavy ion collisions, after mutual capture of colliding nuclei, a moleculelike nuclear configuration (so-called a nuclear molecule or a DNS) is probably to form [5-7]. At this step, nucleons are exchanged between two nuclei as far as the DNS moves to the equilibrium. Before compound nucleus (CN) formation, due to the Coulomb repulsion, the DNS may separate again (about $ {10}^{-21} $$ {10}^{-20} $sec.), which is called quasi-fission, QF [8].

      A nuclear molecule or DNS are two touching nuclei that do motion within the internuclear distance with the transfer of nucleons. So this system consists of two main degrees of freedom: (1) the exchange of nucleons between the projectile and target nuclei and (2) the relative motion between two nuclei, that leading to the fusion-fission and QF respectively. QF and fusion-fission are the severe obstacles to the formation of a superheavy nucleus. The principal difference between QF and fusion-fission is that CN does not form during the QF process [9-11].

      Generally, heavy nuclei are produced in three stages: capture, fusion, and deexcitation. Studying each of these stages can help us understand and analyze the production of heavy nuclei.

      In this article, we want to find an appropriate dinuclear system for the formation of $ {}^{296} $119 superheavy compound nucleus. This nucleus is chosen because the highest known atomic number to date belongs to the Oganesson nucleus, which has 118 protons [12-14]. For this purpose, we consider four reactions for the production of $ {}^{296} $119 superheavy element including: $ ^{45} $Sc + $ ^{251} $Cf, $ ^{42} $Ca + $ ^{254} $Es, $ ^{39} $K + $ ^{257} $Fm, $ ^{38} $Ar + $ ^{258} $Md. These reactions are selected for the following reasons: (1) Projectiles are the stable nuclei. (2) Targets are the nuclei that have an alpha decay. (3) Targets are actinides. (4) Three of the projectiles have a magic number ($ {}^{42}_{20} $Ca, $ {}^{39}_{19} $K, and $ {}^{38}_{18} $Ar) that two of them have a closed neutron shell ($ {}^{39}_{19} $K and $ {}^{38}_{18} $Ar) and one of them has a closed proton shell ($ {}^{42}_{20} $Ca). In other words, all types of nuclei (even-odd, odd-odd, and even-even nuclei) are used in reactions. Selective reactions are debatable only from a theoretical point of view, Because many reactions considered cannot be used due to lack of target material.

      It needs to be said the projectiles heavier than $ {}^{48} $Ca are commonly selected to the synthesis of the superheavy nuclei with proton numbers greater than 118 [15], as in a number of articles [16-22] for production of the element with 119 protons have been suggested two reactions: $ {}^{48} $Ca + $ {}^{252} $Es and $ {}^{50} $Ti + $ {}^{249} $Bk, which leads to the formation of the $ {}^{300} $119 and 299119 nuclei, respectively. But in this article, we want to study the behavior of the projectiles lighter than $ {}^{48} $Ca for the synthesis of a superheavy nucleus with 119 protons.

      This paper reviews the driving potential in the framework of the two-center shell model (TCSM) and folding potential in Section II. The results are presented in Section III, in the form of capture, fusion, and evaporation residue cross section and the survival probability calculations. In the end, the conclusion of our calculations is provided in Section IV.


      A.   Driving potential

    • The potential energy for the heavy nuclear configuration plays a crucial role in our understanding of the evolution of the collision of nuclei. So it is better to start with the driving potential. The potential energy surface which controls the evolution of the nuclear system in multi-dimensional space is commonly called “driving potential” [23]. In other words driving potential included a value of nucleus-nucleus potential corresponding to the minimum of its potential well.

      The potential energy depends on three parameters: (1) the parameter of mass asymmetry ($ \eta $) which is defined as the ratio of the subtraction of the mass of two colliding nuclei to their sum, (2) the distance between mass centers of the colliding nuclei (r), and (3) the neck parameter $ \varepsilon $ [24, 26] which is determined as the ratio of the height of the smoothed potential to the original one [27]. For the fusion reaction, the realistic value of the neck parameter is $ \varepsilon $ = 1.

      The potential energy of the separated nuclei is defined as the interaction energy of the colliding nuclei, which might be calculated using the double-folding method, proximity potential or Bass model (for spherical nuclei). By crossing the fusion barrier the further evolution of the system may be an adiabatic or diabatic process (due to the relative movement speed of the two nuclei) [23, 24]. Using the code in Ref. [24], one can calculate the multi-dimensional diabatic (in the folding model framework) and adiabatic (in the TCSM framework) driving potential. As seen in Fig. 1 diabatic and adiabatic processes must have the same potential before touching of two nuclei, but after touching point these processes have quite different behavior. In the diabatic process, the two nuclei approach each other rapidly, and after contact, try to penetrate each other, and the potential energy increases rapidly. But in the adiabatic process, where the nuclei are slowly approaching each other, the DNS has sufficient opportunity to change its configuration in order to hold nuclear density and the potential energy does not exceed a certain limit. In this paper, we use the adiabatic process for potential driving after touching point. Also, we consider the axially symmetric configurations.

      Figure 1.  (color online) Potential energy as elongation function for the nuclear system formed by (a) $ {}^{45} $Sc + $ {}^{251} $Cf, (b) $ {}^{42} $Ca + $ {}^{254} $Es, (c) $ {}^{39} $K + $ {}^{257} $Fm, and (d) $ {}^{38} $Ar + $ {}^{258} $Md in the framework of TCSM compared with the folding potential.

      In Fig. 1, the TCSM is used in the form of an adiabatic process to consider the shell effect, and the folding potential is used in the form of a diabatic process to consider the effects of deformation and orientation. Their theory is detailed in Sec. IIB and Sec. IIC.

    • B.   TCSM potential

    • The TCSM Hamiltonian in cylindrical coordinates is defined as the sum of potential and kinetic energy which consists of three terms: two-center oscillator, spin-orbit, and $ l^2 $-terms [28, 30]

      $ {\hat{H}}_{\rm TCSM} = -\frac{{\hslash }^2}{2m_0}{\bf{\nabla}}^2+V\left(z,\rho \right)+V_{{LS}}\left({{r}},{{p}},{{s}}\right)+V_{{{L}}^{2}}\left({{r}},{{p}}\right), $


      indicating the places of the centers by $ z_1 $ and $ z_2 $, momentum independent term of the potential

      $ V\left(z,\rho{}\right) = \frac{1}{2m_0}\left\{\begin{array}{l}\left({\omega{}}_{z1}^2{z^{'}}^2+{\omega{}}_{\rho{}1}^2{\rho{}}^2\right),\qquad\qquad\qquad\qquad\qquad\qquad\;\;\; z<z_1 \\ \left[{\omega{}}_{z1}^2{z^{'}}^2\left(1+c_1z^{'}+d_1{z^{'}}^2\right)+{\omega{}}_{\rho{}1}^2{\rho{}}^2\left(1+g_1{z^{'}}^2\right)\right],\qquad z_1<z<0 \\ \left[{\omega{}}_{z2}^2{z^{'}}^2\left(1+c_2z^{'}+d_2{z^{'}}^2\right)+{\omega{}}_{\rho{}2}^2{\rho{}}^2\left(1+g_2{z^{'}}^2\right)\right],\qquad 0<z<z_2 \\ \left({\omega{}}_{z1}^2{z^{'}}^2+{\omega{}}_{\rho{}1}^2{\rho{}}^2\right),\qquad\qquad\qquad\qquad\qquad\qquad\;\;\; z>z_2\end{array}\right. $



      $ z^{'} = \left\{\begin{array}{ll}z-z_1, & z<0 \\ z-z_2,& z>0\end{array}\right. $


      potential of the spin-orbit interaction

      $ V_{LS}\left({{{r}},{{p}},{{s}}}\right) = \left\{\begin{array}{l}\left\{-\dfrac{\hslash{}{\kappa{}}_1}{m_0{\omega{}}_{01}},\left({\bf{\nabla}}V\times{}{{p}}\right)\cdot{{s}}\right\},\ \ \ \ \ z<0 \\ \left\{-\dfrac{\hslash{}{\kappa{}}_2}{m_0{\omega{}}_{02}},\left({\bf{\nabla}}V\times{}{{p}}\right)\cdot {{s}}\right\},\ \ \ \ \ z>0\end{array}\right. $


      and the $ V_{L^2} $ potential is

      $ V_{L^2}\left({{{r}},{{p}}}\right) = \left\{\begin{array}{l}-\dfrac{1}{2}\left\{{\kappa{}}_1{\mu{}}_1{\hslash{}\omega{}}_{01},{{{l}}^{\bf{2}}}\right\}+{\kappa{}}_1{\mu{}}_1{\hslash{}\omega{}}_{01}\dfrac{N_1\left(N_1+3\right)}{2}{\delta{}}_{if},\ \ \ \ \ z<0 \\ -\dfrac{1}{2}\left\{{\kappa{}}_2{\mu{}}_2{\hslash{}\omega{}}_{02},{{{l}}^{\bf{2}}}\right\}+{\kappa{}}_2{\mu{}}_2{\hslash{}\omega{}}_{02}\dfrac{N_2\left(N_2+3\right)}{2}{\delta{}}_{if},\ \ \ \ \ z>0\end{array}\right. $


      In the above formula, $ {\delta }_{if} $ is a Kronecker symbol and $ \left\{x,y\right\}\equiv xy+yx $ states the anticommutator of two quantities. $ {\omega }_{\rho i} $ and $ {\omega }_{zi} $ are the oscillator frequencies in $ \rho $ and z directions, respectively which are $ z_i $-dependent. Also, $ {\kappa }_i $ is the spin-orbit interaction constant, $ {\mu }_i $ is the tunable parameter of the Nilsson model depends on the two-center distance. $ N_i = n_{\rho i}+n_{zi}(z_i) $ states the principal quantum number of the two-center oscillator which $ z_i $-dependent quantity $ n_{zi} $ the solution of a transcendental equation and $ n_{\rho i} $ is a nonnegative integer. $ {\hslash\omega }_{0i} = 41/{\tilde{A}}^{1/3}_i $ is the energy level spacing of the spherical oscillator, where is $ {\tilde{A}}_i $ the mass number of the nuclear fragment [28, 30].

      The potential mentioned in equations (1-5) refers to a single-particle potential. we used the TCSM for the calculation of the adiabatic potential energy of the nucleus-nucleus interaction that can be found in [24].

    • C.   Folding potential

    • The double-folding model is one of the most common methods to determine the internuclear potential for the heavy ion interaction where includes in the sum of the effective interaction of the nucleon-nucleon. The interaction energy of two nuclei in the folding model is calculated as [23, 24, 31]

      $ V_{12}\left(r;{\delta }_1,\Omega _1,{\delta }_2,\Omega _2\right) \!=\! \!\!\int^{\ }_{V_1}{{\rho }_1\left({{r}}_1\right)}\int^{\ }_{V_2}{{\rho }_2\left({{r}}_2\right)v_{NN}\left({{r}}_{12}\right){\rm d}^3{{r}}_1{\rm d}^3{{r}}_2}, $


      in this formula, $ {{r}}_{12} = {{r}}+{{r}}_2-{{r}}_1 $, and $ v_{NN}\left(r_{12}\right) $ shows the effective interaction of the nucleon-nucleon includes two parts: nuclear and Coulomb. Also, $ {\rho{}}_1\left({{r}}_1\right) $ and $ {\rho{}}_2\left({{r}}_2\right) $ are the distributions of the nuclear matter density within the colliding nuclei, which is often calculated by

      $ \rho \left({{r}}\right) = {\rho }_0{\left[1+\exp\left(\frac{r-R\left(\Omega _{{r}}\right)}{a}\right)\right]}^{-1}, $


      $ {\Omega{}}_r $ are the spherical coordinates of r and $ R\left({\Omega{}}_r\right) $ denotes the distance to the nuclear surface. Also, the amount of $ {\rho{}}_0 $ is specified by the condition $\displaystyle\int{\rho{}}_i{\rm d}^3{{r}} = A$. A more detailed description of the folding potential can be found in [31].

      Figure 2 shows the driving potentials as a function of $ \eta $ parameter in the framework of TCSM for the DNS leading to the $ {}^{296}119 $. In this figure it can be seen there are two minimums in the driving potential in the $ \eta = 0.07 $ and $ \eta = 0.4 $ for the reactions at $ R_B $ (derived from Ref. [24]) where minimum potential energy for these minimums are shown in Table 1. This figure also shows the value of the internal fusion barrier $B^*_{\rm fus}$. The important property of the DNS evolution to the CN is the existence of a fusion barrier $B^*_{\rm fus}$ in the mass asymmetry coordinate. The value of the internal fusion barrier determines a hindrance of complete fusion, which DNS must overcome this barrier to form CN [4].

      Reaction $R_B/{\rm fm}$ Mass
      asymmetry ($ \eta $)
      Minimum potential energy/MeV
      The first
      $ \left(\eta{}=0.07\right) $
      The second
      $ \left(\eta{}=0.4\right) $
      $\rm {}^{45}_{\ }{Sc}+{}^{251}_{\ }{Cf}$ 12.716 0.696 179.3 179.7
      $\rm {}^{42}_{\ }{Ca}+{}^{254}_{\ }{Es}$ 12.675 0.716 169.9 170.2
      $\rm {}^{39}_{\ }K+{}^{257}_{\ }{Fm}$ 12.630 0.736 157.3 157.6
      $\rm {}^{38}_{\ }{Ar}+{}^{258}_{\ }{Md}$ 12.622 0.743 153.9 154.3

      Table 1.  The minimum value of the driving potential (in the TCSM) as a function of $ \eta $ parameter in our desired reactions.

      Figure 2.  (color online) Driving potential as the one variable function ($ \eta $) for the DNS leading to the $ {}^{296}119 $ CN formed by (a) $ {}^{45} $Sc + $ {}^{251} $Cf, (b) $ {}^{42} $Ca + $ {}^{254} $Es, (c) $ {}^{39} $K + $ {}^{257} $Fm, and (d) $ {}^{38} $Ar + $ {}^{258} $Md in the framework of TCSM. The red line indicates the entrance channel. Also, the value of the internal fusion barrier for each reaction is shown.

      By using the code in Ref. [24], we can show in Fig. 3 the driving potential as a function of the proton number of the DNS leading to the identical CN with A = 296 and Z = 119 (in contact point). This figure is equivalent to Fig. 2. It is clear from Figs. 2 and 3 that the amount of internal fusion barrier in $ ^{38} $Ar + $ ^{258} $Md reaction is smaller than for other studied reactions.

      Figure 3.  (color online) Driving potential calculated for the DNS leading to the identical CN with A = 296 and Z = 119 as a function of the proton number of the DNS formed by (a) $ {}^{45} $Sc + $ {}^{251} $Cf, (b) $ {}^{42} $Ca + $ {}^{254} $Es, (c) $ {}^{39} $K + $ {}^{257} $Fm, and (d) $ {}^{38} $Ar + $ {}^{258} $Md reaction. The red arrow indicates the initial proton number of projectile for the desired reaction.

      The landscape of the potential energy surface specifies the competition between complete fusion and QF during the evolution of DNS [32]. For this purpose, the dependence of the potential energy on polar orientation for each reaction were studied. In Fig. 4 this dependence on the orientation with the parameter R is observed that shows the parameters of the fusion barriers strongly depend on the orientation of the nuclei during fusion [23, 25, 26]. According to this figure, the minimum and maximum energy occur at $ 0^{\circ} $ (nose-to-nose collision) and $ {\pm 90^{\circ}} $ (side-by-side collision), respectively. Furthermore, as expected maximum energy occurs around the point contact. Minimum energy at the point contact is at $ 0^{\circ} $ for the reactions, where its value is listed in Table 2.

      Reaction Minimum energy ($R_{\rm contact}$)
      $\rm {}^{45}_{\ }{Sc}+{}^{251}_{\ }{Cf}$ 183.48 MeV (12.03$ \pm $0.06 fm)
      $\rm {}^{42}_{\ }{Ca}+{}^{254}_{\ }{Es}$ 181.27 MeV (12.05$ \pm $0.06 fm)
      $\rm {}^{39}_{\ }K+{}^{257}_{\ }{Fm}$ 178.79 MeV (11.95$ \pm $0.06 fm)
      $\rm {}^{38}_{\ }{Ar}+{}^{258}_{\ }{Md}$ 168.34 MeV (11.90$ \pm $0.06 fm)

      Table 2.  The minimum potential energy obtained from the reactions at $ 0 $ degree orientation of the colliding nuclei. point contact is given inside the parentheses.

      Figure 4.  (color online) The Dependence of the potential energy on mutual orientation in the reaction plane and R(distance between mass centers of colliding nuclei) in the (a) $ {}^{45} $Sc + $ {}^{251} $Cf, (b) $ {}^{42} $Ca + $ {}^{254} $Es, (c) $ {}^{39} $K + $ {}^{257} $Fm, and (d) $ {}^{38} $Ar + $ {}^{258} $Md reaction. The mutual orientation is such that $ 0^{\circ} $ and $ \pm90^{\circ} $ angles show nose-to-nose and side-by-side collision, respectively.


      A.   Capture cross section

    • In the fusion of superheavy ions, the CN formation probability after touching of two nuclei is less than unity due to the QF processes. In such systems the cross section of fusion corresponds to the so-called “capture cross section”, which is defined as the QF cross section (without CN formation) plus the fusion cross section (CN formation). The total capture cross section is [3, 23, 25, 26]

      $ {\sigma }_{\rm cap}\left(E,l\right) = \pi {\rlap{-} \lambda }^2\mathop \sum \limits_{l = 0}^\infty {\left(2l+1\right)}T_l\left(E\right), $


      where $ {\rlap{-} \lambda } $ represents the reduced De-Broglie wavelength and $ {\sigma{}}_{cap}\left(E\right) $ describes the transition of two nuclei on the Coulomb barrier with the primary dinuclear system formation and the probability $ T_l\left(E\right) $ when the angular momentum l of the relative motion is converted into the DNS angular momentum in the center-of-mass energy framework. Also as mentioned before, the kinetic energy is converted into the excitation energy, too [33, 34]. The probability of these changes is defined by the Hill-Wheeler equation [35].

      $ T^{\rm HW}_l\left(B,E\right) = {\left(1+\exp\left(\frac{2\pi }{\hslash{\omega }_B\left(l\right)}\left[B+\frac{\hslash^2}{2\mu R^2_B\left(l\right)}l\left(l+1\right)-E\right]\right)\right)}^{-1}, $


      which $ \hslash{\omega }_B = \sqrt{\hslash^2/\mu {\left|{\partial }^2V/\partial r^2\right|}_B} $ represents the potential barrier width, B denotes the barrier height and $ R_B\left(l\right) $ represents the position of the effective barrier contains a centrifugal part. The penetration probability for spherical nuclei is average over B [24-26]:

      $ T_l\left(E\right) = \int{F}\left(B\right)T_l^{HW}\left[B(\beta);E\right]{\rm d}B, $


      which $ F(B) $ can be approximated by symmetric Gaussian [24-26]

      $ F(B) = N.\exp{\left(-\left[\frac{B-B_{0}}{\Delta_B}\right]^{2}\right)}, $


      where $ B_0 = (B_{1}+B_{2})/2 $ and $ {\Delta_B} = (B_{2}-B_{1})/2 $. The quantity $ B_1 $ depended on dynamic deformation and $ B_2 $ is the Coulomb barrier of spherical nuclei.

      For our studied reactions which consists of the spherical projectile and statically deformed target, the penetration probability must be averaged on the deformation-dependent barrier height also the colliding nuclei orientations. The Coulomb barrier height dependent on the dynamic deformation of the projectile. Therefore probability of penetration is as follows [24]

      $ T_l\left(E\right) = \int_0^{\pi{}}\frac{\sin{\theta{}}_2}{2}{\rm d}{\theta{}}_2\int{F}\left(B^{'}\right)T_l^{\rm HW}\left[B(\theta,\beta);E\right]{\rm d}B^{'}, $


      Whit considering ($ \theta $) as a target orientation and ($ \beta $) as a projectile deformation, the relationship between B and $ B^{'} $ to parameterize the barrier B for the arbitrary value of the ($ \theta $) and ($ \beta $) is as follows [24]:

      $ B\left(\theta{},\beta{}\right) = B^{'}+\left[B\left(\theta{},0\right),B\left(0,0\right)\right],\ \ \ \ B^{'} = B\left(0,\beta{}\right). $


      The effective nucleus-nucleus potential which can be seen inside the Formula Eq. (9) Brackets is approximated around the Coulomb barrier by the potential of the inverted harmonic oscillator with the frequency $ \omega (l)$ and the maximum amount of angular momentum ($l_{\max}$) is determined by either the kinematical angular momentum as $l_{\rm kin} = {\left\{2\mu \left[E_{\rm c.m.}-V\left(R_b,\ Z_i,\ A_i,\ {\beta }_1 = 0,\ {\beta }_2 = 0,\ l\right)\right]\right\}}^{{1}/{2}}{R_b}/{\hslash}$ or by the critical angular momentum: $l_{\max}$ = minimum$ {\{} $$l_{\rm kin}$, $l_{\rm cr}$$ {\}} $ [36]. The used value of the critical angular momentum for our reactions in this study is specified in Table 3.

      reactions critical angular momentum
      $\rm {}^{45}_{\ }{Sc}+{}^{251}_{\ }{Cf}$ 121
      $\rm {}^{42}_{\ }{Ca}+{}^{254}_{\ }{Es}$ 119
      $\rm {}^{39}_{\ }K+{}^{257}_{\ }{Fm}$ 116
      $\rm {}^{38}_{\ }{Ar}+{}^{258}_{\ }{Md}$ 116

      Table 3.  The used amount of the critical angular momentum in the studied reactions.

      In Fig. 5 cross section of the fusion and capture is calculated. The difference between these cross sections is due to the absence of the CN in the QF process [9, 10].

      Figure 5.  (color online) Fusion (solid curve) and Capture (dashed curve) cross sections calculated for our reactions: (a) $ {}^{45}_{21} $Sc + $ {}^{251}_{98} $Cf, (b) $ {}^{42}_{20} $Ca + $ {}^{254}_{99} $Es, (c) $ {}^{39}_{19} $K + $ {}^{257}_{100} $Fm, and (d) $ {}^{38}_{18} $Ar + $ {}^{258}_{101} $Md.

    • B.   Fusion cross section

    • Cross section of the CN (fusion) is calculated by [16]

      $ {\sigma }_{\rm fus}\left(E,l\right) = \frac{\pi {\hslash }^2}{2\mu E}\mathop \sum \limits_{l = 0}^\infty {\left(2l+1\right)}T_l\left(E\right)P_{\rm CN}, $


      $P_{\rm CN}$ is the fusion probability that leads to the compound nucleus formation. The difference between ${\sigma}_{\rm fus}$ and ${\sigma}_{\rm cap}$ is in a $P_{\rm CN}$ coefficient which generally considered in computation such that the CN with a probability of 100% is $P_{\rm CN} = 1$. Thus in a heavy system, the capture process within the Coulomb barrier or DNS formation is not enough condition for fusion but is the necessary condition for fusion [33, 34]. A simple parameterization of $P_{\rm CN}$ was proposed in [37, 38]

      $ P_{\rm CN}\left(E,l\right) = \dfrac{\exp(-c(x_{\rm eff}-x_{\rm thr}))}{1+\exp\left(\dfrac{E^*_B-E^*_{\rm int}\left(l\right)}{\Delta}\right)}, $


      the excitation energy of CN (at the Bass barrier) denoted by $ E^*_B $. Also, $E^*_{\rm int}\left(l\right) = E-E_{\rm rot}\left(l\right)+Q$ is the inner excitation energy, where describes the damping of the shell correction to the CN fission barrier. $ \Delta $ is the tunable parameter of about 4 MeV and Q is the fusion Q-value and $E_{\rm rot}\left(l\right) = \dfrac{{\hslash }^2}{2{\mathfrak{I}}_{g.s.}}l\left(l+1\right)$ is the rotational energy [16, 23, 25]. Also, c, $x_{\rm eff}$, and $x_{\rm thr}$ can be found in [37, 38].

      According to Table 4. not only maximum values for ${\sigma}_{\rm fus}$ and ${\sigma}_{\rm cap}$ in the reaction $ {}^{38} $Ar + $ ^{258} $Md are greater than other reactions studied, but also this reaction requires lower energy. It can be said as the mass asymmetry parameter ($ \eta $) increases, maximum values for the fusion and capture cross section increases as well (see Table 2. and Table 4.).

      Reaction Maximum capture cross section ($E_{\rm cm}$) Maximum fusion cross section ($E_{\rm cm}$)
      $\rm {}^{45}_{\ }{Sc}+{}^{251}_{\ }{Cf}$ 855.234 mb (275$ \pm $2.5 MeV) 0.859 mb (265$ \pm $2.5 MeV)
      $\rm {}^{42}_{\ }{Ca}+{}^{254}_{\ }{Es}$ 861.806 mb (255$ \pm $2.5 MeV) 5.921 mb (260$ \pm $2.5 MeV)
      $\rm {}^{39}_{\ }K+{}^{257}_{\ }{Fm}$ 954.991 mb (255$ \pm $2.5 MeV) 39.891 mb (255$ \pm $2.5 MeV)
      $\rm {}^{38}_{\ }{Ar}+{}^{258}_{\ }{Md}$ 1005.70 mb (245$ \pm $2.5 MeV) 200.349 mb (245$ \pm $2.5 MeV)

      Table 4.  Maximum value for ${\sigma }_{\rm cap}$ and ${\sigma }_{\rm fus}$ in this study.

    • C.   Evaporation residue cross section

    • Formation cross section of evaporation residue for heavy nuclei collisions given as follows [39]

      $ {\sigma }_{\rm EvR}\left(E_{\rm c.m.}\right) = \mathop \sum \limits_{J = 0}^{{J_{\max}}} {{\sigma }_{\rm cap}\left(E_{\rm c.m.},J\right)P_{\rm CN}\left(E_{\rm c.m.},J\right)W_{\rm sur}\left(E_{\rm c.m.},J\right)}, $


      where three coefficients include: cross section of capture, complete fusion probability, and survival probability. The cross section of capture ${\sigma{}}_{\rm cap}$ describes the DNS formation in the first step of collision when kinetic energy (due to the relative movement) is converted into other energies (excitation energy and potential energy). Once formed, the DNS evolves within the coordinate of the mass asymmetry $ \eta $. The mass distribution center moves in the direction of greater symmetric fragmentations. If part of mass distribution passes through the internal fusion barrier $B_{\rm fus}^*$ of driving potential $ U\left(\eta{}\right) $, it gives a probability of complete fusion $P_{\rm CN}$. Also during its evolution, the DNS can decay with QF. Hence, the charge and mass distributions of the QF and probability of the fusion $P_{\rm CN}$ should behave at the same time [39].

      The excitation energy of the formed compound nucleus is about $ E^* $ = 30-40 MeV (or $E_{\rm cm}$ = 180-190 MeV), and the transition of the excited compound nucleus to the ground state results in the emission of 3 or 4 neutrons from its surface [40]. Figure 6 shows the calculated evaporation residue cross sections for the 3n and 4n channels in the combinations studied. It is clear that values of the $ {\sigma }_{EvR} $ in the $ {}^{38} $Ar + $ {}^{258} $Md reaction for both channels are greater than other reactions and this is a good sign that distinguishes this reaction from the reactions mentioned for the production of $ {}^{296} $119 superheavy element.

      Figure 6.  (color online) (a) Cross sections of the evaporation residue calculated for the 3n channel and (b) for the 4n channel in the reactions.

    • D.   The survival probability for the excited CN

    • For an excited CN, the probability to achieve a ground state with the emission of a neutron (or neutrons) is defined as “survival probability” $W_{\rm sur}$ [39]. This probability computes the contest among particle evaporation in the $ xn, yp, z\alpha $ channel, with the excited CN fission and other particle evaporation channels [41, 42]. Therefore the total survival cross section is the sum of the survival probability over all channels:

      $ {\sigma}_{\rm surv}\left(E,l\right) \!=\! \pi {\rlap{-} \lambda }^2 \!\!\mathop \sum \limits_{l = 0}^\infty \!\!{\left(2l+1\right)}T_l\left(E\right)P_{\rm CN}\left(E,l\right)\!\!\sum\limits_{xn,yp,z\alpha}\!\!W^{xn,yp,z\alpha}_{\rm surv}\!\!\left(E,l\right), $


      Using the following equation the survival probability for the 1n-evaporation channel is obtained:

      $ W^{1n}_{\rm sur}(E_{\rm c.m.},J = 0) = \frac{\Gamma_n\left(E^*_{\rm CN}\right)}{\Gamma_f\left(E^*_{\rm CN}\right)} \\ = \frac{4A^{{2}/{3}}\left(E^*_{\rm CN}-B_n\right)}{k\left(2{\left[a\left(E^*_{\rm CN}-B_f\right)\right]}^{{1}/{2}}-1\right)}\exp{\left[2a^{{1}/{2}}\left(\sqrt{E^*_{\rm CN}-B_n}-\sqrt{E^*_{\rm CN}-B_f}\right)\right]}, $


      where k = 9.8 MeV and $E^*_{\rm CN} = E_{\rm c.m.}+Q$. Also, $ a = (A_1+ A_2)/{12} $ is the ratio for the level density which its value in the evaporation channels is equal to one. $ B_f $ is the fission barrier for the heaviest nuclei, where its amount proportional to the CN excitation energy ($E_{\rm c.m.}$) as $B_f = B_f\left(E^*_{\rm CN} \!=\! 0\right)\exp\left[-E^*_{\rm CN}/E_d\right]$, which $E_d \!=\! 0.4A^{4/3}/a$ [3, 33, 34].

      Studies show that the probability of the survival strongly proportional to the properties of the nuclear structure for the superheavy nuclei like deformation and level density [31], which we have applied these properties in our calculations.

      Results of calculation for $W_{\rm sur}$ and $ {\sigma }_{EvR} $ in the 1-4n channels (in the range of 180-200 MeV) for the $ {}^{38}_{18} $Ar + $ {}^{258}_{101} $Md reaction by using codes in Ref. [24] are shown in Fig. 7. According to this figure, maximum value for $ {\sigma }_{EvR} $ at the 3n-channel and 4n-channel are 0.18(pb) and 0.05(pb), respectively, which is a good agreement with the results of others [16-22] for the production of an element with 119 protons. The various results may be related to the various methods, potentials, energy range, and neutron number of the element.

      Figure 7.  (color online) The survival cross section (dashed curve) and cross section of the evaporation residue (color curves) calculated for the 1n-4n channels in $ {}^{38}_{18} $Ar + $ {}^{258}_{101} $Md reaction.

    • We used several codes available in Ref. [24] to study the formation of $ {}^{296} $119 superheavy compound nucleus at energies near the Coulomb barrier. Our calculations and studies lead to the following conclusions:

      For the reactions with a larger mass asymmetry parameter, the minimum potential value in the contact point is further reduced. $ {}^{42}_{20} $Ca, $ {}^{39}_{19} $K, and $ {}^{38}_{18} $Ar, which are spherical and magic nuclei have more amount than $ {}^{45}_{21} $Sc in the capture (even fusion) cross section and survival probability. For $ {}^{39}_{19} $K and $ {}^{38}_{18} $Ar which have closed neutron shells, the survival probability for CN is greater than other studied reactions without closed neutron shells; but $ {}^{42}_{20} $Ca which has closed proton shells, the CN survival probability is less than nuclei with closed neutron shells. Also as the mass asymmetry parameter ($ \eta $) increases, maximum values for the capture and fusion cross section increases as well.

      Although maximum values for the capture and fusion cross sections are higher in the reaction $ {}^{38} $Ar + $ {}^{258} $Md, it is not precise to conclude that this reaction is the optimal one among mentioned four combinations just based on the fusion cross sections, because the excitation energy of the compound nucleus could be much higher, which suppresses the survival probability. At the same time, this reaction is a suitable choice for the formation of an element with 119 protons among our studied reactions, because the calculated evaporation residue cross section in $ {}^{38} $Ar + $ {}^{258} $Md reaction is larger than other reactions and also this reaction has a smaller fusion barrier.

Reference (42)



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