Fusion Reaction Study of some Selected Halo Systems

: The challenge in studying fusion reaction when the projectile is neutron or proton rich halo nuclei is the coupling mechanism between the elastic and the breakup channel, therefore the motivation from the present calculations is to estimate the best coupling parameter to introduce the effect of coupled-channels for the calculations of the total cross section of the fusion 𝝈 𝒇𝒖𝒔 , the barrier distribution of the fusion 𝑫 𝒇𝒖𝒔 and the average angular momentum 〈 L 〉 for the systems 6 He+ 206 Pb, 8 B+ 28 Si, 11 Be+ 209 Bi, 17 F+ 208 Pb, 6 He+ 238 U, 8 He+ 197 Au and 15 C+ 232 Th using quantum mechanical approach. A quantum Coupled-Channel Calculations are performed using CC code. The predictions of quantum mechanical approach are comparable with the measured data that is available. Above and below the Coulomb barrier, comparison of theoretical calculations of quantum mechanical with the relevant measured data demonstrates good agreement.


Introduction:
Investigation of the mechanisms of the reaction involving the collisions of weakly bound nuclei at the energies near to the barriers which are stable or radioactive is one of the main fields of Low Energy Nuclear Physics (LENP). Comprehensive theoretical and experimental studies have been focused on understanding the various processes that occur in a collision, as well as the significant couplings that exist between them. On this topic, a few global review papers have been written 1 . The breakup channel plays an important role in collisions of the systems that are tightly bound. Transfer channels are essential, particularly for neutron-halo nuclei. The cross sections have large breakup for the systems of the weakly bound have due to their low breakup threshold, and the breakup mechanism has efficient on other reaction channels 2 . The study of such reactions is significant in astrophysics, and several of those reactions could be significant gateways for super-heavy elements production. Fusion is affected by two in a weakly bound projectile collision. firstly, a nature of the static, the fusion barrier has various property relative to the fusion barrier for the same target for a tightly bound isotope. The barriers are lower due to the longer density tail. At the energies near to the sub-barrier, it is generally thought to contribute to greater fusion cross sections. The dynamic of the elastic channel coupling with inelastic, breakup, and transition channels is the second thing to consider. The bound states inelastic excitations and the channels of the direct transfer have been shown to enhance the cross sections of the fusion at the energies near to sub-barrier. On the contrary, the break-up method has a number of features when it establishes continuum status, and the impact of connections to sources is debatable in the continuum. As the binding is weak, the breakup is still quite wide and the elastic channel is very strongly coupled. Thus, the cross section, along with the normal transfer and inelastic channels, can play a relevant role 3 . The role of the breakup channel has been studied extensively by Majeed and his collaborators and for more details we refer the readers to refs. [4][5][6][7][8] . Fusion reaction for some light, medium and heavy mass systems using the semiclassical approach seems very competitive to study the fusion reactions properties [9][10][11][12][13] .
The purpose of this work is to employ quantum mechanics by means of the coupledchannels to study the effect of coupling the breakup channel on the theoretical predictions of the total cross section of the fusion , the barrier distribution of the fusion and the average angular momentum 〈L〉 for the systems 6   where 0 belongs to the depth of potential and 0 was the potential diffuseness. The radius of the nuclear potential was described as with 14 , where the angle is between ⃗ , and and are the polynomials of Legendre. Due to the potential, the solution enhanced in behaviour. Because the potential reaches zero at infinity, the wave-function asymptotic shape can be described similar to Eq. 2. the plane waves will be replacing with the corresponding waves of Coulomb, the form of asymptotic is 16 Fusion reactions could be considered as incident flow absorption. The s-matrix absolute value is less than unity, when the potentials are complex, i.e. | | < 1, and from Eq. 3 It is possible to quantify the difference in total radial flux between the incoming and outgoing waves 18 To derive Eq. 11, for all the values of to be considered, the radial flux has been integrated and divided by the incident flux, = ℏ ⁄ the fusion cross section is then taken as 18 In the heavy-ion fusion, the regular boundary condition at origin is not imposed Eq. 8, IWBC is called the "incoming wave boundary condition" and it has been applied with the goal of preserving the real potential. The wave function has a form under this boundary condition and can be written as 18,19 ( . . . .13 at a distance less than radius absorption r , which is considered to be within the barrier of Coulomb. ( ) is number of the local the wave for the -th wave partial, which can be described as 18 2 2 ) . . . . . . . 14 Due to the occurring of the strong absorption in the region of the interior, the incoming wave boundary condition is equal such that the incoming flux is not returned. The final results for heavy-ion fusion reactions are not really dependent on the radius of absorption r and the minimum position of the potential is also considered. With IWBC, in Eq. 13 is the coefficient of transmission, therefore the S-matrix, S in Eq.7 is nothing but the coefficient of reflection. So, Eq.14 is turned into 18,20 ( ) = 2 ∑ (2 + 1) (

The distribution of the fusion barrier
The classical cross section of the fusion can be written as

Results and Discussion:
In this work, the theatrical calculations of the cross-section of fusion , the distribution barrier and mean angular momentum 〈 〉 using the quantum mechanical approach for the systems 6 He+ 206 Pb, 8 B+ 28 Si, 11 B+ 209 Bi and 17 F+ 208 Pb, 6 He+ 197 Au, 6 He+ 238 U, 8 He+ 197 Au and 15 C+ 232 Th. The quantum mechanical calculations using CC code for the , and 〈 〉 are compared with measured data. The parameters of the potential (Woods-Saxon potential) with different parametrizations which is called Winther-Akÿz potential used throughout in the current calculations are illustrated in Table 1. These parameters were obtained by the least-square fit method to match the correct position of the experimental barrier height+ at the center point.  The results of the quantum mechanical calculations show in Fig. 1 in the panels (a, b, c) for , b and 〈 〉, respectively, for the system 6 He+ 206 Pb. The results show that the coupling calculations and non-coupling channels were able to describe the measured data below and around Coulomb barrier . The calculations and measured data 22 in all figures are plotted in both logarithm and linear scales for (mb). The calculations enhanced markedly below the Coulomb when the coupling channel were included. For this system the average angular momentum 〈 〉 agrees well with the measured data for both coupling and 30.8/*without coupling. Fig.2 (a, b and c) depicted the comparison of theory and measured data of , and a 〈 〉 with the measured data for the system 8 B+ 28 Si. The measured data are taken from 23 . Our results of for both coupling and no coupling channels reproduce the measured data, whereas the linear curves for coupling and no coupling channels results overshoot the measured data. The results of our calculations for , the coupling channel predictions agreed more reasonably with experimental data 22 , while the results of the mean angular momentum 〈 〉 overshoots the measured data above the Coulomb barrier, for both coupling and no-coupling. Fig.3 panel (a), (b) and (c) compared theoretical results with measured data for , and 〈 〉 for the system 11 Be+ 209 Bi. The calculations of both logarithmic and linear curves for coupling and no coupling channels results overshoot the measured data 24 . The results of the calculations with channel coupling are in agreed more with the measured data for the , while the coupling channel result for average angular momentum 〈 〉 agrees with the measured data above the barrier. Fig.4 depicts the comparison of the calculated , and 〈 〉 with their corresponding measured data for both coupling and no coupling for the system 17 F+ 208 pb. The calculations of for both coupling and noncoupling could not describe two experimental data 25 below the Coulomb barrier Vb, because 17 F is proton rich halo nucleus that made suppression to the fusion cross section which makes theoretical predictions overshoots the measured data. Including the coupling in the calculations of reproduce the behaviour of the experimental data and deviate from the Gaussian shape in the case of no coupling. The calculations with coupling included for 〈 〉 agrees with experiment above Vb and overshoots the experimental values below Vb. Fig.5 panel (a), (b) and (c) show the results of the quantum mechanical calculations for , and 〈 〉, respectively, for the system 6 He+ 238 U compared to the experimental data 25 . The results show that calculations included coupling and no-coupling channels were able to describe the measured data below and around Coulomb barrier . The inclusion of the coupled channel enhances the calculations of below the Coulomb barrier. While and 〈 〉 calculations with shows ripple behaviour with two peaks appear for which are closer to the experimental data. Fig.6 panel (a), (b) and (c) presented the theoretical predictions compared to measured data 26 for , and 〈 〉 for the system 8 He+ 197 Au. The location of barrier of the measured data for Coulomb appear as an arrow on the . . axis. For this system the inclusion of the coupling has very slight effect on the peak height of the calculations and have same results in the case of nocoupling and in general have no agreement with the experimental data for the and 〈 〉 calculations expect very slightly difference in the peak value of .
The calculated results of , and 〈 〉 were compared with the measured data 27 in panels (a), (b) and (c) of Fig.7, respectively for 15 C+ 232 Th system. The inclusion of coupled channel in quantum mechanical calculations enhanced markedly the calculations for the below the Coulomb barrier makes theoretical predictions in more match with the measured data and were able to reproduce the data satisfactorily. The quantum mechanical with and without coupling couldn't describe the experimental data for the and 〈 〉. The results of the calculations for fusion cross section for both coupling and no-coupling channels agreed with the measured data in both logarithmic and linear curves. The results of the calculations for the coupling channel results overshoot the measured data for this reaction. For mean angular momentum 〈 〉 the coupling channel

Conclusion:
The conducted study detailed theoretical investigation of the calculations for , and t 〈 〉 for the systems 6 He+ 206 Pb, 8 B+ 28 Si, 11 Be+ 209 Bi, 17 F+ 208 Pb, 6 He+ 238 U, 8 He+ 197 Au and 15 C+ 232 Th were analyzed and discussed. The best fitted coupling parameters were obtained based on the best fitted parameters of the Woods-Saxon parameters that match the centroid of the measured and calculated barrier height Vb. The results compared to experiment reveals that the breakup channel is very necessary to describe , , and 〈 〉 for these systems involving light halo nuclei as a projectile. A comparison of our theoretical calculations of quantum mechanical with the measured data display a reasonable agreement with the available measured data. The results of this study are very important for researchers who are interested in studying the physics of weakly bound systems where the projectile is a halo nucleus.