Coupled Channels Calculations of Fusion Reactions for 46 Ti + 64 Ni , 40 Ca + 194 Pt and 40 Ar + 148 Sm Systems

In this work, the fusion cross section ,  fusion barrier distribution  and the probability of fusion  have been investigated by coupled channel method  for the systems 46Ti+64Ni, 40Ca+194Pt and 40Ar+148Sm with semi-classical and quantum mechanical approach using SCF and CCFULL Fortran codes respectively. The results for these calculations are compared with available experimental data. The results show that the quantum calculations agree better with experimental data, especially bellow the Coulomb barrier, for the studied systems while above this barrier, the two codes reproduce the data.


Introduction
In the processes of the colliding nuclei, when the two separate nuclei are overcoming the Coulomb barrier, and fuse together to produce a compound nucleus, this process is known as fusion reactions. The Coulomb barrier is a resultant of a repulsion Coulomb and attractive nuclear forces. At energies below this barrier (classically forbidden region), fusion can take place by tunneling phenomenon. The tunneling probability or transmission coefficient can be evaluated by Wentzel-Kramers-Brillouin (WKB) method [1]. For light nuclei, the probability of fusion can be described by this barrier with the radial motion between nuclei which represent the only degree of freedom. But medium and heavy nuclei, have internal degrees of freedom and coupling between these and transitional motion can enhances the fusion cross section below the barrier [2][3][4][5]. This coupling can by represented by taking into account the rotational deformations, vibrational modes or nucleons transfer processes [1,6]. The best method to investigate this coupling is the solution of coupled channel equations which are performed with semi-classical approach of Alder and Winther (AW) method for Continuum Discretized Coupled Channel (CDCC) where generalized from Coulomb excitations to nuclear reactions [7].
In the present work, the CDCC method was adopted to evaluate the fusion cross section, fusion barrier distribution and the probability of fusion for the medium systems 46 Ti+ 64 Ni, 40 Ca+ 194 Pt and 40 Ar+ 148 Sm, with SCF code, recently adopted by [8]. The results of this approach were compared with full quantum mechanical calculations which were performed by CCFULL code and recently been adopted by [5], in addition to the available experimental data for these systems.

Theoretical framework
Evaluate the fusion cross section in the semi-classical coupled channels model, was done by solving AD equations for CDCC method. This procedure was performed by researchers [9][10][11][12]. The for all channels ( ) is given by: is the wave number, ℓ represents the orbital quantum number and the probability of fusion is ℓ ( ), which is expressed by [10,12,13] ( ) = ∫ | ( , )| ( ) (2) where ℓ ( , ) is the radial wave equation for the ℓ th partial wave in channel , E represents the center of mass energy and ( ) is the imaginary part of the optical potential associated to fusion in this channel. In the quantum mechanical calculation, the for no-Coriolis approximation is given by [1]: where J is the total angular momentum. In the coupling, the barrier can be thought as a set of subbarriers or barrier distribution, and the fusion occurs by these subbarriers. The fusion barrier distribution is an important tool to investigate the mechanism of fusion and nuclear structure of colliding nuclei [1,3,14]. Rowley et al. found the expression to evaluate this function by the experimental data of [15].
which was found theoretically from by three point difference method [3,16] At energy (E 1 +2E 2 +E 3 )/4 and for equal spacing between the values of energy (∆ ), the above equation can be written as The second derivative statistical error is give as [16] where is the absolute cross section uncertainties.

Results and discussion
The semi-classical coupled channels calculations for the systems 46 Ti+ 64 Ni, 40 Ca+ 194 Pt and 40 Ar+ 148 Sm have been performed by SCF code to evaluate fusion cross section, fusion barrier distribution and the probability of fusion. These results were compared with the available experimental data and full quantum mechanical calculations which were carried out by CCFULL code with two modes of excitation both in projectile and target nuclei.
The real parameters of nuclear potential, Wood-Saxon potential, for the above systems which were obtained by fitting the experimental data of fusion cross sections are listed in Table 1, where ° is the potential depth, ° is the radius constant and diffuseness °.

The 46 Ti+ 64 Ni system
In the quantum mechanical calculations, the vibrational coupling with single phonon for Ti and Ni was taken into account. The deformation parameters corresponding to multipolarity λ are listed in Table 2. The , and results are shown in Fig.1 panels a, b and c, respectively. The measured data (green circles) were taken for this system from [17]. Below the Coulomb barrier V b , as indicated by the (magenta arrow on the E c.m. axis) the calculations of quantum mechanics (the red curve) for , performed by (CCFULL) are in better agreement with the measured data, as shown in Fig.1 panel (a), while the semi-classical results (black curve), which were accomplished by SCF code are shortfall the data. Above V b , despite the results of CCFLL code are closest to the data, but the results of SCF code for are close as well.
For D fus calculations, Fig.1 panel (b), the results of CCFULL show two peaks of barriers around V b while only one peak appears in SCF results. These peaks in quantum calculations gives an enhanced sub-barrier fusion as shown in panel (a), especially below V b . Fig.1.c illustrates the results of probability of fusion, below V b , quantum mechanical results are closer to the data than the results of SCF code, for the same reason above. Above V b , the results of calculations are matching the experimental data.
This enhancement in quantum calculations below the barrier of fusion is due to the coupling to the vibrational excitations, which included low lying states 2 + and 3 − for participant nuclei, leading to contribution of these channels in the reaction.  Fig. 2 (a, b and c) represent the , and for this system. The experimental data were taken from [20]. The quantum mechanical calculations were performed with two modes of vibrational excitation of projectile and single mode of vibrational excitation of target with two phonons for both projectile and target nuclei. The deformation parameters for this system are listed in Table 2. As shown in Fig.2(a), the results of CCFULL code for (red curve) with low lying states 2 + and 3 − for Ca nucleus and 2 + state for Pt nucleus with two phonons for both nuclei at energies below V b are closer to experimental data than the results of SCF code, as shown in Fig.2(a). While above V b , the two curves are coincide with the experimental data. As shown in the Fig. 2(a), although the results of CCFULL code are better than that of the SCF code below the barrier, but it is still less than the data at these energies. This reduction in calculations below the fusion barrier indicates the presence of other channels of interaction below this barrier. When adding the two-neutron

The 40 Ca+ 194 Pt system
pickup channel to the excitation due to vibration channel, a significant enhancement in the results below the barrier were noticed, as shown in the blue curve in this figure. Ground state energy of neutron transfer channel for this reaction Q gg is equal to 5.23MeV and the configuration factor was arbitrarily chosen to be (0.9) to obtain this enhancement.
The results of fusion barrier distribution Fig. 2(b), shows a spectrum of barriers around V b for the quantum calculations, (red curve), while there was only a single barrier for the semi-classical calculations. These results explain the improvement in for the quantum calculations. The results of CCFULL code was obtained with vibrational excitations which show more than one barrier. These barriers correspond to different channels of interaction which did not appear in the semi-classical calculations as these calculations do not include the interaction channels resulting from the deformations of the interacting nuclei.
The probability of fusion, Fig.2(c), shows that the results of the quantum calculation well match the experimental data below V b , while at energies above V b the two curves are well match the experimental data.

The 40 Ar+ 148 Sm system
The results of , and calculations for this system shows in Fig. 3(a, b  and c). Two modes of vibrational coupling for colliding nuclei with single phonon is adopted in this system. The parameters of deformation are listed in Table 2. The results of , Fig. 3(a) shows a better agreement for both CCFULL and SCF codes with the experimental data which were taken from [21]. In quantum calculations, the coupling of low-lying states of 2 + and 3 − levels were adopted which corresponding to quadrupole and octupole vibration in nuclei.
The D fus of this system, Fig. 3(b) shows that the red curve of CCFULL code has god fit with the experimental data than the black curve of SCF code. Fig.3(c), represents the probability of fusion, the two curves are coincide and well represent the data.

Conclusions
In this study, the results of the full quantum mechanics for , and , are better than the semi-classical results especially below the Coulomb barrier of the studied systems. Whereas, the effect of introducing the deformation parameters for the colliding nuclei, as well as the coupling to vibrational states and neutrons transfer processes, in the quantum calculations led to a remarkable enhancement in the results of these calculations below the fusion barrier. While above this barrier, the semiclassical and the quantum calculations reproduced the experimental data for these systems.