STUDY OF FUSION REACTIONS OF LIGHT PROJECTILES ON LIGHT AND MEDIUM TARGETS †

The fusion and breakup reactions of some light projectiles on light and medium targets using semi-classical and full quantum mechanical approaches were adopted to calculate the total cross section 𝜎 (cid:3033)(cid:3048)(cid:3046) and the distribution of the fusion barrier 𝐷 (cid:3033)(cid:3048)(cid:3046) for the systems 12 C + 48 Ti , 16 O+ 63 Cu, 35 Cl+ 25 Mg and 35 Cl+ 27 Al. The coupling between the channel’s contribution from elastic and breakup channels were considered to show their importance in the calculations. The results compared with the measured data and shows reasonable matching, and it is shown that the coupling considered is very essential to be considered, especially below the Coulomb barrier V b .


INTRODUCTION
One of the most important modern researches fields in nuclear physics is studying the collision of weakly bound stable and radioactive nuclei, around the potential barrier [1].The understanding of the processes associated with these reactions can be achieved throw adopting both theoretical and experimental offers in order to obtain the best agreement between them.many researches during the last years supported the strong relation between the different nuclear reaction modes starting by the elastic scattering and ending with the fusion reaction, so that will provide the researchers with a wide field of reaction modes to collect more information about the secrets of the nuclear structure and properties of our universe ingredients.The weakly bound systems collisions are very influenced by both transfer and breakup channels which have a very large cross-section according to their low breakup threshold.The projectile mass is one of the most effective factors on the reaction strength, so it is very important to study the reactions with medium mass projectile to examine the relationship between reaction modes and bombarding energy and nucleon number.The fusion of weakly bound colliding projectiles was very effected statically by their fusion barrier characteristics such as its long tail energy which cause a lower barrier and by the way will give an enhancement to the cross section of the fusion at the energies at the sub-barrier [2,3].Dynamically, fusion was affected by the different channels coupling, including the elastic, breakup, inelastic and transfer ones [4].The kinetic energy of the projectile should exceed the corresponding Coulomb barrier for producing nuclear reaction [5].
All of the energy, mass number, momentum, charge, spin and parity are conserved during the nuclear reaction.The Q-value have a great effect on the fusion calculations because it represents the amount of energy emitted or absorbed during the reaction [6].The two colliding nuclei in the fusion reaction are considered as objects with rigid spherical shape that interact with the potential barrier so the probability of nuclear fusion to accurse represents the ability of the system to penetrate the potential barrier [7], 8].There are many important factors that have a major role in the experimental determination of the fusion cross section such as the colliding nuclei internal degrees of freedom relative motion, the particles transfer and the nuclear deformation [9,10].The collision process is very complex at low temperature, so to be understood it need to unify the description for the different reaction mechanism with a unique nuclear potential [11].The fusion is very complex reaction because of the combination between the coulomb and nuclear interactions in addition to the effect of the flexible intrinsic synthesis during the reaction and the different reaction channels [12].
This study aims to study the fusion reaction of light projectiles on light targets for the systems 12 C + 48 Ti , 16 O+ 63 Cu, 35 Cl+ 25 Mg and 35 Cl + 27 Al by using semi-classically and full quantum mechanically methods where the coupling between the elastic and breakup channels will be considered.

THE SEMI-CLASSICAL TREATMENT
2.1.The single channel theory In one-dimensional potential model, we need to use the semiclassical approach for the fusion cross-section determination by eliminating the degree of freedom by the relative motion between the colliding heavy ion only.[13,14,15].Semi-classically, this can be treated by the assumption of energy and momentum independent Schrödinger equation; where  and ⩗  are the reduced mass and total energy potential of the system respectively.The time dependence function can be used to determine the semiclassical amplitudes by evaluating the particle trajectory using classical dynamics, including all the potential types, as; In addition, the complex potential which represents the imaginary part of the nuclear potential, should be contained.
=   −   . ( The above method can be used to study the effect of the nuclear potential with its real and imaginary parts on the interacting  waves [13,16,17].According to the semi-classical theory, the fusion takes place when the nuclei be closer to the barrier, and the WKB approximation can be used to determine the penetration probability below the barrier [13,18,19,20].
Then it can be simplified as: Where  ( ) and  ( ) represent the turning points of the fusion barrier potential for its inner and outer and  () is the wave number.If a parabolic function used as an approximation for the fusion barrier, then the Hill-Wheeler formula can be used to find the penetration probability above the barrier [13].
is the bombarding energy and  () is the height parameter of the partial wave fusion barrier with curvature parameter Ω .The fusion cross-section can be determined by using the WKB approximations as [17,21]: In the above equation, u γl ( γ ,) refers to the wave function of the radial part in γ channel, and the potential imaginary part denoted by  () .The using of semiclassical theory to compute heavy ions fusion cross section by approximating the trajectory , and the projectile intrinsic states ( ƹ ) using the Coupled-Channel Continuum Discretized (CCCD) method with the helpful of Winther and Alder (AW) theory [21,22,23,24,25].The Hamiltonian of the projectile is, where h (ƹ) is the Hamiltonian fundamental states and ⩗ (ƹ, ) is the interaction potential that determine as; The path of Rutherford transmits on the reaction energy,, and the momentum, .Classically, the potential can be solved as; where Ψ refers to the bounded state of the projectile.Therefore, time dependence Schrödinger equation have been satisfied inn both ξ-space ⩗ (ƹ.) =   ( ) , ƹ and Hamiltonian for intrinsic eigenstates |ψ [26,27], The wavefunction expansion as a function of the intrinsic ground sate is, Then the AW equation can be written as; The AW equations can be computed by the assumption of the ground state at initial conditions  (,  → −∞) =  .

The Coupled Channel Description
The dynamics of the projectile-target can be described by using  ⃗ and  in the projectile intrinsic Hamiltonian  () and the interaction of the projectile-target ( ⃗, ) as; [28], The eigenstates of  () is [28], is the internal motion energy.
There are two steps to consider the AW method.First, the evolution of time of the variable  ⃗ has been considered classically.The energy , and the momentum ℏℓ are the two effected parameters on the path with ( ⃗) = ⟨ |( ⃗, )| ⟩, where | ⟩ represents the ground level.The coupling will be a time dependence and  ℓ (, ) ≡ ( ⃗ ℓ (), ).Second, the quantum mechanical time-dependent problem has been used to treat the dynamics in the intrinsic apace.Throw expanding the wavefunction as [29], the AW equations can be evaluated by substituting the above expansion into Schrodinger equation, we get [32], Under the initial conditions at the ground state the solution of the coupled differential equations can to be obtained by assuming , where ℓ is the angular momentum at  channel.the integration of the cross section gives [29,30], For a simple determination of the fusion reaction cross section, the whole contribution channels can be assumed to be bound to zero spin.Using the expansion of the wave function for all contributions, leads to [18], with, Where  is the imaginary part of the optical potential in the channel  and  ℓ ( , ) its ℓth-partial wavefunction.
The approximated formula can be adopted to find the cross section with the help full of AW, as [28], above,  ℓ ( ) represents the probability of  -channel for the system to be at classical trajectory, and  ℓ ( )  is the probability for the particle at  =  −  and reduced mass  =   /( +  ) , referring to the masses of the projectile and target by  ,  , respectively, [28].By using loosely bound projectiles, the CF for some systems will be studied.For simplicity, the projectile ground state is considered to be the only bound one in which the breakup reaction achieved in two parts, F 1 and F 2 .therefore, it will be referred to the ground and breakup states by the labels  = 0 and  ≠ 0 , respectively.If the sequential contribution has been neglected, the CF can only contribute from the elastic channel.So,  is determined as, ℓ is the survival probability, which is given by; [18].

Quantum Mechanical Approximation
The relative motion between the colliding nuclei in addition to the nuclear intrinsic degrees of freedom need to be studied quantum mechanically by assuming Ψ(, ) to be the entire wave function for the reaction with r represents the separation vector of the projectile and target while  refers to their intrinsic coordinates set.By the Hamiltonian, the reaction dynamics can be determined as [18],  =  +  +  (23).

EEJP. 2 (2023) Malik S. Mehemed
In which H 0 represents the inherent Hamiltonian, T is the operator of the energy associated with the collide nuclei movement which given as  ≡ −ℏ  /2, and the potential of the interaction  ≡ (, ).an intrinsic Hamiltonian with eigenstates |⟩ can satisfy the Schrödinger equation as [13], With where the wave function  ()  () is corresponding to |⟩ ( |⟩) state in the space.The potential represented as, where  is the channel space diagonal, such that [13], , (r) =   * ()  (r, )  (). ( The potential  is not random for the diagonal in channel space.It is appropriate to take  in the case of  is not diagonal and  =  −  with [13], from the Schrödinger equation, the equation of the coupling is, and the expansion, where |Ψ(  )⟩ represents the collision initiating in channel  ,  is the wave vector, the energy scale was chosen to be  = 0 .The Schrödinger equation solution components according to the off-diagonal part of the reaction are |Ψ (  )⟩ for  =  and  ≠  .The Hamiltonian written as [13], We can get the coupled channel equations from Eqs. ( 24), ( 25) and ( 23) as, [13], With using | (  )⟩ ⟶  () in Eq. ( 25), we get, The imaginary part  refers to the flux gain by the other channels from channel .The non-Hermitian nature of H leads to break down of the continuity equation, while for the Hermitian  in the coupled channel interaction, the continuity equation has the form [30].
represents the probability density.By taking the integration of Eq. (39) in spherical region covering the interaction area and with the helpful of  definition, we get [22], the potential absorption is given as; where  refers to the lost flux and  refers to the fusion absorption in channel α, the total cross section represented as [22], Cl+ 25 Mg and 35 Cl + 27 Al .The WS potential parameters are tabularized in Table 1.
Table 1.The WS potential parameters for the studied systems 4.1 12 C + 48 Ti System The obtained  and  for 12 C + 48 Ti are drown in Figure 1 with its labels (a) and (b), respectively.The semiclassical calculations are represented in blue colour curves, while quantum mechanical calculations are represented in red colour curves.The solid and dashed curves represent the calculations with and without the channel coupling respectively.Figure 1 show that the best obtained calculations for both  and  under and above the Coulomb barrier V b are those including the channel coupling in the quantum mechanical calculations.

16 O+ 63 Cu System
The calculations for  are more accurate for those treated using the simiclassical treatment with channel coupling as shown in panel (a) of Figure 2, while the best calculations for  are those treated using the quantum mechanical treatment with channel coupling as shown in panel (b) of Figure 2.

CONCLUSION
The results for all the studied systems show a remarkable influence for the channel coupling on the calculations of  and  for 12 C + 48 Ti, 16 O+ 63 Cu, 35 Cl+ 25 Mg and 35 Cl + 27 Al systems, also we conclude that quantum mechanical treatment was proved to be successful for the total cross section determination while the fusion barrier distribution.The semi-classical calculations succeeded in describing the measured data especially above the Coulomb barrier V b .

Figure 1 .
Figure 1.The semiclassical calculations with the blue colour and quantum mechanical calculations with the red colour for both  and  in panels (a) and (b) respectively for the system 12 C + 48 Ti

Figure 2 .
Figure 2. The semiclassical and quantum mechanical calculations for both  and  in panels (a) and (b) respectively for the system 16 O+ 63 Cu 4.3 35 Cl+ 25 Mg System The calculations for  are in more agreement with the experimental data for those treated using the semi-classical approach with channel coupling as shown in panel (a) of figure 3, while the best  calculations are those treated using the channel coupling in quantum mechanical treatment as shown in panel (b) of the figure.

Figure 3 .
Figure 3.The semiclassical and quantum mechanical calculations for both  and  in panels (a) and (b) respectively for the system 35 Cl+ 25 Mg 4.4 35 Cl + 27 Al System Panel (a) in figure4show in panel (a) that the best obtained calculations for  under and above the Coulomb barrier V b are those calculated using semi-classical treatment with effect of coupled channel included, while the panel (b) show that the best calculations for  are those obtained using the quantum mechanical treatment with the effect of channel coupling.

Figure 4 .
Figure 4.The semiclassical and quantum mechanical calculations for both  and  in panels (a) and (b) respectively for the system 35 Cl + 27 Al