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 <sup>6</sup> Li + 64Ni, 11B + 159Tb, and 12C + <sup>9</sup> Be. 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 [3–8]. 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, 8–10]. 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 <sup>4</sup> He+<sup>2</sup> H, with separation energy Sα = 1.48 MeV; 11B breakup into

The method can be extended to describe interference of different l waves due to strong nuclear attraction and absorption caused by the imaginary nuclear potential [6, 20, 22]. When the two nuclei come across the potential barrier into the inner region, the fusion occurs according to the semiclassical theory, and the penetrability probability below barrier can be evaluated usingWKB approximation [5, 6, 19, 23, 24]:

<sup>1</sup> <sup>þ</sup> exp 2 <sup>Ð</sup> <sup>r</sup>

<sup>1</sup> <sup>þ</sup> exp <sup>2</sup><sup>π</sup>

<sup>1</sup> <sup>þ</sup> exp <sup>2</sup><sup>π</sup>

where Vbð Þl and Ω<sup>l</sup> are the height and the curvature parameter of the fusion barrier for the partial wave, respectively, and E is the bombarding energy. Ignoring

<sup>κ</sup><sup>2</sup> <sup>∑</sup>ð Þ <sup>2</sup><sup>l</sup> <sup>þ</sup> <sup>1</sup> <sup>P</sup>WKB

where u<sup>γ</sup><sup>l</sup> k<sup>γ</sup> ;r � � represents the radial wave function for the partial wave l in

The complete fusion cross section in heavy ions evaluated using semiclassical theory is based on the classical trajectory approximation r. And the relevant intrinsic degrees of freedom of the projectile, represented by ξ with applying the continuum discretized coupled-channel (CDCC) approximation of Alder and Winther (AW) theory [16], have been proposed [17]. The projectile Hamiltonian is then

where h0ð Þξ is the intrinsic Hamiltonian and Vð Þ¼ ξ;r VNð Þþ ξ;r VCð Þ ξ;r , Vð Þ ξ;r is the interaction between the projectile and target nuclei. The Rutherford trajectory depends on the collision energy, E, and the angular momentum, l. In this case the trajectory is the solution of the classical motion equations with the potential

dr u<sup>γ</sup><sup>l</sup> <sup>k</sup><sup>γ</sup> ;<sup>r</sup> � � � � �

� 2 W<sup>γ</sup>

h ¼ h0ð Þþ ξ Vð Þ ξ;r (9)

fusð Þr is the absolute value of the imaginary part of the potential

classical turning points at the fusion barrier potential. If one approximates the fusion barrier by a parabolic function, then the penetrability probability above

ℏΩ<sup>l</sup>

ð Þl <sup>b</sup> and r

ℏΩ<sup>l</sup>

ð Þl a r ð Þl b

ð Þ VbðÞ�l E

ð Þl

ð Þ E � Vbð Þl

<sup>κ</sup>lð Þ<sup>r</sup> dr � � (4)

h i (5)

h i (6)

fus ð Þ l; E (7)

fusð Þr (8)

<sup>a</sup> are the inner and outer

fus ð Þ¼ <sup>l</sup>; <sup>E</sup> <sup>1</sup>

fus ð Þ¼ <sup>l</sup>; <sup>E</sup> <sup>1</sup>

fus ð Þ¼ <sup>l</sup>; <sup>E</sup> <sup>1</sup>

the l dependence of ω and of the barrier position Rb and assuming that the l dependence of Vbð Þl is given only by the difference of the centrifugal potential energy, one can obtain Wong's formula which is given in Section 5. The fusion cross sections can be estimated by the one-dimensional WKB approximation by the

<sup>σ</sup>fusð Þ¼ <sup>E</sup> <sup>π</sup>

fusð Þ¼ <sup>l</sup>; <sup>E</sup> <sup>4</sup><sup>k</sup>

E ð

PWKB

Fusion Reaction of Weakly Bound Nuclei DOI: http://dx.doi.org/10.5772/intechopen.80582

Then, the latter can be rewritten as follows:

PWKB

where κlð Þr is the local wave number and r

barrier is given by the Hill-Wheeler formula [3, 19]:

PWH

Pγ

associated to fusion in that channel.

following relations [21, 24]:

channel γ and W<sup>γ</sup>

given by

23

4 He+<sup>7</sup> Li with separation energy Sα = 8.664 MeV and 12C breakup into three α particles induced by neutrons or protons by 12C (p, p<sup>0</sup> ) 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 σfus and fusion barrier distribution Dfus for 6,8H halo [17]. Semiclassical coupled-channel calculations in heavy-ion fusion reactions for the systems 40Ar + 110Pd and 132Sn + 48Ca 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 σfus and the fusion barrier distribution Dfus calculations quite well below and above the Coulomb barrier for medium and heavy systems. This study aims to employ a semiclassical approach by adopting Alder and Winther (AW) [19] theory originally used to treat the Coulomb excitation of nuclei. The semiclassical approach has been implemented and coded using FORTRAN programming codenamed (SCF) which is written and developed by Canto et al. [20]. The fusion cross section σfus and fusion barrier distribution Dfus are calculated here utilizing the semiclassical approach. The results from the present study are compared with the quantum mechanical calculations using the FORTRAN code (CC) [21] and with the experimental data for the three systems <sup>6</sup> Li + 64Ni, 11B + 159Tb, and 12C + <sup>9</sup> Be.
