Numerical Analysis of Fusion Cross Section of O 16 + O 16 by Using The Modified Glas-Mosel Formula

One of the interesting topics in nuclear reactions is a study about reaction cross section between the interacting nuclei. For calculating fusion cross section, the Glas-Mosel formula has been proven successfully in explaining the experimental results of fusion cross section. In this study, the fusion cross sections of O 16 + O 16 reaction were calculated by using modified Glas-Mosel formula. The energies were set at 10 ≤ E ≤ 40 MeV. The potential of interacting nuclei was approached by using Woods-Saxon potential. In numerical process, the differential equations were solved by using finite different method and optimization process was performed by using Nelder-Mead method. Good agreement between the experimental and this study results has been achieved successfully. Referring those results above, it can be indicated that the modified GlasMosel formula has good capability to explain the experimental results of fusion reaction of light nuclei. It can be a useful tool in explaining the experimental results or in predicting fusion cross section of light nuclei.


INTRODUCTION
One of the interesting topics that should be concerned in order to study about nuclear reaction, especially in charged-particle nuclear reactions, is the height of the barrier between the interacting nuclei is.It can provide information on the fusion process by a measurement.It is an important intermediate step to produce super-heavy nuclei by heavy-ion reactions.
In 1974, Glas and Mosel [1] proposed a model to analyze both high and low energies fusion cross section data for heavy-ion collisions.It has been proven successfully in explaining the experimental results by other researchers [2,3] .
In this study, it was calculated the fusion cross section of O 16 + O 16 reaction at 10 ≤  ≤ 40 MeV of energies by using modified Glas-Mosel formula.The calculation results have been also compared with the experimental results obtained by Fernandez et al. [4] .In addition, the calculation results of O 16 + O 16 reaction obtained by the previous researchers, with different methods, have been also displayed.Simenel et al. [5] worked with the densityconstrained time-dependent Hartree-Fock (DC-TDHF) method to explain the experimental results.Kondo et al. [6] used the optical model with the deep ion-ion potential to calculate the fusion cross sections.
Another purpose of this study is to observe the capability of modified Glas-Mosel formula and its precision in predicting the fusion cross section.It is very important information as a fundamental reference for further study about fusion cross section using heavy and superheavy nuclei.It is also very useful to provide fusion cross section data, especially for unavailable experiment data or the difficulty in cross section measurement by experiments.

METHOD
In this study, it is considered first in calculation fusion cross sections that the interacting nuclei are spherical and without dynamical distortion.The various barriers are approximated for different partial waves by inverted harmonic-oscillator potentials of height   and frequency   .For an energy , the probability (, ) for the absorption of the -th partial wave is then given by the Hill-Wheeler formula [7]  , = {1 + exp ( where ℏ is the reduced Planck constant and   represents the height of barrier potential, well known as Coulomb barrier (MeV).The total potential can be calculated by using [8,9]   ( where  represents the atomic number,  is the electron charge,  0 represents the potential depth,  respresents the distance between the interacting nuclei,  0 represents the radius, and  represents the surface diffuseness parameter.The parameters of  0 ,   , and   can be estimated by using the following approximations [8,9,10]  0 = 16 ̅ where  0 represents the radius parameter,   represents the critical distance, and   represents the parameter of critical distance.The constants of  and  0 were set respectively at 1.8 and 0.95 (in MeVfm −2 ) [8] .The subscript  and  refer to projectile and target respectively.The parameter of ℏ  (where ℏ  ≃ ℏ  ), represents the characterizing the behavior of the fusion cross section at very low energy near and below the Coulomb barrier, which can be approached by where  is the reduced mass of nuclei.The parameter ℏ is the (-independent) curvature of the parabolic barrier assumed in the calculation.Finally, the fusion cross sections can be calculated by using Wong formula [7,9]   ( In other hand, in the phenomenological model of Glas and Mosel, the fusion cross section is given by [1]   ( where   (  ) represents the barrier (critical) distance and   represents the critical potential.The parameter   is obtained as the peak of total potential.
In this study, it is proposed a modified Glas-Mosel formula as where  1 is the parameter.The calculations were performed numerically.The numerical differential was calculated by using finite different method, which can be formulated as [11]  2 where ∆ represents the step of axis.For optimizing the parameters of ,  0 ,   , and  1 , it was utilized Nelder-Mead method.This method is very useful to optimize computational problems by using numerical method or to solve the analytic problems with unknown gradient [12] .The capability of this method has been proven successfully in solving the function with two variation parameters [12,13] .
The step of this method is not so complicated.For example, consider (, ) as function.
The first step, it is taken three trial coordinates, i.e. ( 1 ,  1 ), ( 2 ,  2 ), and ( 3 ,  3 ).After calculated, those coordinats are noted as follows:  = ( 1 ,  1 ) as the best vertex (vertex with minimum value),  = ( 1 ,  1 ) as good vertex, dan  = ( 1 ,  1 ) as the worst vertex.Next step, the function is optimized by following the algorithm explained in Table 1.Many examples in optimizing the functions can be found in the following reference [12] .
Table 1.The algorithm of Nelder-Mead method. [12]se (i) To ensure the precision of calculation, chi-square distribution can be utilized as long as the experimental data can be extracted.The chi-square can be calculated by using [14,15] 2   (15)   where Δ  exp represents the uncertainty in measurements and  is the number of data.The minimum value of  2 measures the quality of the fit.The smaller the value of  2 , the higher the quality of the fit [14] .

RESULTS AND DISCUSSION
Calculation process in this study is commenced by optimizing ,  0 ,   , and  1 in Nelder-Mead framework.These parameters were optimized in order to get minimum chi-square of fusion cross section.To guarantee the saturation of calculation, the chi-square tolerance was set at 10 −20 .The obtained results of this process are:  = 0.6282 fm,  0 = 1.2072 fm,   = 1.1822 fm, and  1 = 0.7012 with  2 = 0.4106.It can be seen that the obtained chi-square is less than 0.5 which means that the excellent agrement between this calculation and the experimental result has been achieved successfully.
Next step after optimization, it is calculated the fusion cross sections for O 16 + O 16 reaction by using the modified Glas-Mosel formula, as shown in eq. ( 13).The calculation results of some parameters obtained inthis study are shown in Table 2.The potential models used in this study have been displayed in Figure 1.It can be seen from Figure 1 that, at distance above 6 fm, Woods-Saxon potential moves in approaching zero exponentially., obtained by Fernandez et al. [4] , Simenel et al. [5] , Kondo et al. [6] , Thomas et al. [16] , and this study.
The modification of Glas-Mosel formula proposed in this study has reduced the fusion section value obtained by using Wong formula, especially at 15 ≤  ≤ 40 MeV of energies.This study results is also close to the calculation results of Kondo et al.It means that, in using the modified Glas-Mosel formula, when the condition of  ≥   is achieved, the fusion cross section values decrease.This decrease is influenced by the denominator term of log function of eq. ( 13).Therefore, good agreement between this study results and the experimental results has been achieved successfully for O 16 + O 16 reaction.
Based on those results above, the modified Glas-Mosel formula, with the obtained parameters, proposed in this study, has good capability in explaining the experimental results of fusion cross sections of O 16 + O 16 .It can be inferred that the modified Glas-Mosel formula can be a useful tool in explaining the experimental results or in predicting fusion cross section of light nuclei in order to provide the nuclear data.In other hand, it should be applied to the heavy nuclei, even super-heavy nuclei, to investigate the capability of this formula.

Figure 2 2 .
Figure 2 shows the calculation results of fusion cross sections obtained in this study.It has been also compared to the experimental results obtained by Fernandez et al. and to the calculation results of Simenel et al. and Kondo et al.As comparison, it is also calculated the fusion cross section by using Wong formula as shown in eq.(11).The results of this study by using Wong formula are higher than those of Simenel et al.By using the modified Glas-Mosel formula, the results of this study are almost similar to those of experiment.

Table 2 .
Calculation results of   +   reaction.