The fusion barrier parameters are height

## Abstract

Semiclassical and full quantum mechanical approaches are used to study the effect of channel coupling on the calculations of the total fusion reaction cross section σfus and the fusion barrier distribution Dfus for the systems 6Li + 64Ni, 11B + 159Tb, and 12C + 9Be. The semiclassical approach used in the present work is based on the method of the Alder and Winther for Coulomb excitation. Full quantum coupled-channel calculations are carried out using CCFULL code with all order coupling in comparison with our semiclassical approach. The semiclassical calculations agree remarkably with the full quantum mechanical calculations. The results obtained from our semiclassical calculations are compared with the available experimental data and with full quantum coupled-channel calculations. The comparison with the experimental data shows that the full quantum coupled channels are better than semiclassical approach in the calculations of the total fusion cross section σfus and the fusion barrier distribution Dfus.

### Keywords

- fusion reaction
- breakup channel
- weakly bound nuclei
- fusion barrier

## 1. Introduction

In recent years, big theoretical and experimental efforts had been dedicated to expertise the effect of breakup of weakly bound nuclei on fusion cross sections [1, 2]. This subject attracts special interests for researchers and scholars, because the fusion of very weakly bound nuclei and exotic radioactive nuclei is reactions that have special interests in astrophysics which play a very vital role in formation of superheavy isotopes for future applications [3, 4, 5, 6, 7, 8]. Since the breakup is very important to be considered in the fusion reaction of weakly bound nuclei, the following should be considered: the elastic breakup (EBU) in which neither of the fragments is captured by the target; incomplete fusion reaction (ICF), which happens when one of the fragments, is captured by the target; and complete fusion following BU (CFBU), which happens in all breakup fragments that are captured by the target, is called the sequential complete fusion (SCF). Therefore, the total breakup cross section is the sum of three contributions: EBU, ICF, and CFBU, whereas the sum of complete fusion (including two body fusions and CFBU) and incomplete fusion is called total fusion (TF) [1, 8, 9, 10]. Fusion reactions with high-intensity stable beams which have a significant breakup probability are good references for testing the models of breakup and fusion currently being developed. The light nuclei such as ^{6}Li breakup into ^{4}He+^{2}H, with separation energy Sα = 1.48 MeV; 11B breakup into ^{4}He+^{7}Li with separation energy Sα = 8.664 MeV and ^{12}C breakup into three α particles induced by neutrons or protons by ^{12}C (p, p′) 3α [3, 11, 12]. The breakup channel is described by the continuum discretized coupled-channel (CDCC) method. The continuum that describes the breakup channel is discretized into bins [13, 14]. To study the coupled-channel problem, this requires a profound truncation of the continuum into discrete bin of energy into equally spaced states. The CDCC method is totally based on this concept. Surrey group extended the discretization procedure discussed in [14] for the deuteron case to study the breakup and fusion reactions of systems involving weakly bound nuclei [15, 16]. Recently, Majeed and Abdul-Hussien [17] utilized the semiclassical approach based on the theory of Alder and Winther. They carried out their calculations to investigate the role of the breakup channel on the fusion cross section ^{6,8}H halo [17]. Semiclassical coupled-channel calculations in heavy-ion fusion reactions for the systems ^{40}Ar + ^{110}Pd and ^{132}Sn + ^{48}Ca were carried out by Majeed et al. [18]. They argued that including the channel coupling between the elastic channel and the continuum enhances the fusion reaction cross section ^{6}Li + ^{64}Ni, ^{11}B + ^{159}Tb, and ^{12}C + ^{9}Be.

## 2. The semiclassical theory

### 2.1 The single-channel description

The semiclassical theory is used to estimate the fusion cross section in the one-dimensional potential model which assumes that one can describe the degree of freedom only of the relative motion between the colliding heavy ions [19, 20]. The semiclassical theory deals with the Schrödinger equation assuming independent energy and angular momentum and the potential energy for the radial part of the relative motion through quantum tunneling:

where

In coupled-channel effects on the elastic channel, the imaginary part should be added to the nuclear potential, represented by complex potential as

The method can be extended to describe interference of different

Then, the latter can be rewritten as follows:

where

where

where

The complete fusion cross section in heavy ions evaluated using semiclassical theory is based on the classical trajectory approximation

where

After expanding the wave function in the basis of intrinsic eigenstates

and inserting Eq. (11) into the Schrödinger equation for

These equations should be solved with initial conditions

### 2.2 The coupled-channel description

The variables employed to describe the collision are the projectile-target separation vector

where

where

and inserting this expansion into the Schrödinger equation for

These equations are solved with the initial conditions

To extend this method to fusion reactions, we start with the quantum mechanical calculation of the fusion cross section in a coupled-channel problem. For simplicity, we assume that all channels are bound and have zero spin. The fusion cross section is a sum of contributions from each channel. Carrying out partial-wave expansions, we get [28]

with

Above,

To use the AW method to evaluate the fusion cross section, we make the approximation [27]

where

We now proceed to study the CF cross sections in reactions induced by weakly bound projectiles. For simplicity, we assume that the g.s. is the only bound state of the projectile and that the breakup process produces only two fragments,

where

is usually called survival (to breakup) probability [19].

## 3. Fusion barrier distribution

Nuclear fusion is related to the transmission of the incident wave through a potential barrier, resulting from nuclear attraction plus Coulomb repulsion. However, the meaning of the fusion barrier depends on the description of the collision. Coupled-channel calculations include static barriers, corresponding to frozen densities of the projectile and the target. Its most dramatic consequence is the enhancement of the total fusion reaction cross section

where

The experimental determination of the fusion reaction barrier distribution has led to significant progress in the understanding of fusion reaction. This comes about because, as already mentioned, the fusion reaction barrier distribution gives information on the coupling channels appearing in the collision. However, we note from Eq. (24) that, since fusion reaction barrier distribution should be extracted from the values of the total fusion reaction cross section, it is the subject to experimental as well as numerical uncertainties [29, 30, 31]:

where

where

## 4. Results and discussion

In this section, the theoretical calculations are obtained for total fusion reaction ^{6}Li + ^{64}Ni, ^{11}B + ^{159}Tb and ^{12}C + ^{9}Be. The values of the height

### 4.1 The reaction ^{6}Li + ^{64}Ni

The calculations of the fusion cross section ^{6}Li + ^{64}Ni. The dashed blue and red curves represent the semiclassical and full quantum mechanical calculations without coupling, respectively. The solid blue and red curves are the calculations including the coupling effects for the semiclassical and full quantum mechanical calculations, respectively. Panel (a) shows the comparison between our semiclassical and full quantum mechanical calculations with the respective experimental data (solid circles).

The experimental data for this system are obtained from Ref. [32]. The real and imaginary Akyüz-Winther potential parameters obtained by using chi-square method are the strength

### 4.2 The reaction ^{11}B + ^{159}Tb

In similar analysis we compare our theoretical calculations of the fusion cross section ^{11}B + ^{159}Tb. The experimental data for this system is obtained from Ref. [33]. The real and imaginary Akyüz-Winther potential parameters are obtained by using chi-square method:

### 4.3 The reaction ^{12}C + ^{9}Be

Figure 3 (panels (a) and (b)) presents the comparison between our theoretical calculations for ^{12}C + ^{9}Be. The experimental data for this system are obtained from Ref. [34]. The real and imaginary Akyüz-Winther potential parameters are obtained by using chi-square method:

## 5. Conclusion

The semiclassical and quantum mechanical calculations for the total fusion reaction ^{6}Li + ^{64}Ni, ^{11}B + ^{159}Tb, and ^{12}C + ^{9}Be. We conclude that the breakup channel is very important to be taken into consideration to describe the total fusion reaction

## Acknowledgments

The author FA Majeed gratefully acknowledges financial assistance from Conselho Nacional de Desenvolvimento Científico e Tecnolόgico (CNPq) (Brazil) and is especially indebted to the World Academy of Sciences for the advancement of science in developing countries (TWAS) (Italy) for a 1-year grant under the scheme (TWAS-CNPq exchange programs for postdoctoral researchers).