Analysis of 9 Be Fusion Cross Sections via a Simple Cluster Model

The effects of different cluster configurations of 9 Be nucleus on the cross-sections of 9 Be + 28 Si, 9 Be + 64 Zn, 9 Be + 144 Sm, 9 Be + 186 W and 9 Be + 208 Pb fusion reactions have been explored for the first time using a simple cluster approach. The real potential has been calculated based on the α + α + n, d + 7 Li, 3 H + 6 Li, 3 He + 6 He and n + 8 Be cluster cases of the 9 Be nucleus while the imaginary potential is evaluated as Woods-Saxon potential. It has been seen that our results are in agreement with the experimental data. In addition to this, the fusion barrier height (V B ) and barrier position (R B ) values have been given for each reaction and cluster case.


Introduction
A cluster structure can be evaluated as a structure resulting from the movements of nucleons in a nucleus. Thus, a structure consisting of nucleons can be thought as one body [1]. In this respect, the 9 Be nucleus can be assumed as α + α + n, d + 7 Li, 3 H + 6 Li, 3 He + 6 He and n + 8 Be cluster structures [2][3][4] although the cluster structure of 9 Be is not known exactly yet [5]. Therefore, cluster structure is still a hot topic, and many studies can be found in the literature.
In recent years, Aygun has proposed a simple cluster method, and has performed the elastic scattering calculations of 9 Li [6], 9 Be [7], 12 Be [8], 12 B [9] and 22 Ne [10] nuclei. He has gotten the results compatible with the experimental data. However, this approach has not been applied to fusion cross-sections (FCSs).
Therefore, we believe that it will be useful to We examine the effects of different cluster configurations of the 9 Be nucleus on the FCSs of 9 Be + 28 Si, 9 Be + 64 Zn, 9 Be + 144 Sm, 9 Be + 186 W and 9 Be + 208 Pb reactions over a simple cluster model. 9 Be with low binding energy and cluster structure is an important nucleus in the field of nuclear physics. Also, it has a usage area in the field of nuclear technology like thermonuclear devices [11,12]. We first obtain the density distributions for α + α + n, d + 7 Li, 3 H + 6 Li, 3 He + 6 He and n + 8

Calculation Process
The total effective potential that is a significant parameter in the analysis of fusion reactions can be considered as The real potential is gotten for five cluster structures of 9 Be. Detailed information about these structures can be found in our previous study [7]. The calculations of the real potential are carried within the framework of the DF potential shown by ( ) where γ1, γ2, γ3 and γ4 are 7999 MeV, 4.0 fm −1 , 2134 MeV and 2.5 fm −1 , respectively. The imaginary potential is applied in the Woods- where W0, rw and aw are the depth, radius, and diffuseness parameters, respectively. The codes DFPOT [19] and FRESCO [20] are used in the DF model and the cross-section calculations, respectively.

Results and Discussion
The total potential for the theoretical analysis of the fusion reactions has been calculated using Equation (1). According to this, the real potential has been acquired via the DF model by using the density distributions calculated for five different cluster cases of 9 Be. The imaginary potential has been taken as the WS potential. The appropriate values of WS potential have been researched and given in Table 2. Thus, we have plotted the total potential according to the distance in Figure 1. Table 2. The imaginary potential parameters (W0, rw and aw) used in the theoretical analysis of 9 Be + 28 Si, 64 Zn, 144 Sm, 186 W and 208 Pb fusion reactions for α + α + n, d + 7 Li, 3 H + 6 Li, 3 He + 6 He and n + 8 7 Li, 3 H + 6 Li, 3 He + 6 He and n + 8

Be cluster cases
As will be seen from Figure 1, the deepest potential for all reactions has been obtained for α + α + n cluster case. Accordingly, it means that this potential is more attractive than the potentials of the other cluster cases. In addition, it has been seen that the shallowest potential has been gotten for n + 8 Be. Finally, it has been observed that nuclear pocket width has increased as the nucleon number of target nucleus has increased.
Then, we have calculated the FCSs of 9 Be + 28 Si reaction for α + α + n, d + 7 Li, 3 H + 6 Li, 3 He + 6 He and n + 8 Be cluster cases of 9 Be. We have compared the results with the data in Figure 2.
We have tracked that the α + α + n and 3 H + 6 Li   9 Be + 28 Si reaction calculated using α + α + n, d + 7 Li, 3 H + 6 Li, 3 He + 6 He and n + 8 Be cluster cases. The data is from [21] We have obtained the FCSs of 9 Be + 64 Zn reaction for the same cluster structures. We have compared the theoretical results with the data in Figure 3. We have realized that the results are generally similar to each other, and display good consistent with the data.  Figure 2, but for 9 Be + 64 Zn fusion reaction. The data is from [22] Then, we have acquired the FCSs of 9 Be + 144 Sm reaction for five different cluster cases.
We have showed our results together with the data in Figure 4. We have monitored that the cluster results are different from each other at small angles while they are very close at further angles. Additionally, we can say that their compatibility with the data is good. Figure 4. The same Figure 2, but for 9 Be + 144 Sm fusion reaction. The data is from [23] We have gotten the FCSs of 9 Be + 186 W reaction for various cluster cases of the 9 Be nucleus. We have compared the results with the data in Figure 5. We have experienced that the cluster cases are very close to each other at forward angles, and are in good agreement with the data in general sense. Additionally, we have observed that n + 8 Be cluster case is slightly better than the other cluster cases. Figure 5. The same Figure 2, but for 9 Be + 186 W fusion reaction. The data is from [24] Finally, we have calculated the FCSs of 9 Be + 208 Pb reaction for α + α + n, d + 7 Li, 3 H + 6 Li, 3 He + 6 He and n + 8 Be cluster cases of 9 Be. We have compared our results and the data in Figure 6. We have observed that the d + 7 Li and 3 H + 6 Li cluster cases are very close to each other. Also, we have noticed that other cluster results, except for n + 8 Be cluster case, have shown an average behavior with the data.
However, we have observed that the compatibility of n + 8 Be result with the data is very good, and is much better than the other cluster results.

Aygun
Sinop Uni J Nat Sci 6(1): 33-41 (2021) ISSN: 2536-4383 Figure 6. The same Figure 2, but for 9 Be + 208 Pb fusion reaction. The data is from [25] In the present study, we have also calculated the fusion barrier height (VB) and barrier position (RB) for each reaction and each cluster case.
They are known as basic parameters in defining nuclear fusion reactions. We have listed all the values of VB and RB in Table 3. We have obtained the highest RB value and the lowest VB value for n + 8 Be cluster case in all the reactions.
In addition to this, we have found for α + α + n cluster case the highest VB value except for 9 Be + 28 Si and 9 Be + 64 Zn reactions and the lowest RB value in all the reactions. Thus, it can be said that the kinetic energy of the projectile for n + 8 Be cluster state can be less than the other cluster states.

Conclusions
We have calculated the FCSs of 9 Be + 28 Si, 9 Be + 64 Zn, 9 Be + 144 Sm, 9 Be + 186 W and 9 Be + 208 Pb reactions for the α + α + n, d + 7 Li, 3 H + 6 Li, 3 He + 6 He and n + 8 Be cluster structures of 9 Be. We have compared the theoretical results with the experimental data. We have obtained agreement results with the data. We have also given the fusion barrier height and barrier position for each reaction and each cluster case.
Consequently, we have provided new results on various fusion reactions of 9 Be over a simple cluster approach. We think that this approach will be useful to apply to other fusion reactions.

Acknowledgments -
Funding/Financial Disclosure The author has no received any financial support for the research, or publication of this study.

Conflict of Interests -.
Authors Contribution Author read and approved the final manuscript.