THE EFFECT OF THE PROTON AND NEUTRON AS PROBE FOR THE NUCLEAR FUSION REACTIONS AT NEAR-BARRIER ENERGIES †

In this study, quantum mechanical calculations and a semi-classical approach were used to determine fusion the probability (P fus ), fusion barrier distribution (D fus ), and fusion cross section ( σ fus ) for the systems 28 Si + 90 Zr, 28 Si + 92 Zr, 28 Si + 94 Zr, 41 K + 28 Si, and 45 K + 28 Si. The semi-classical approach involved the use of the Wentzel–Kramers–Brillouin (WKB) approximation to describe the relative motion between the projectile and target nuclei, and the Continuum Discretized Coupled Channel (CDCC) method of Alder-Winther (AW) to describe the intrinsic motion of the nuclei. The results showed that the consideration of the coupling-channel calculations for quantum mechanics and a semi-classical approach, and its impact on P fus , D fus , and σ fus for the studied systems involving one neutron or one proton transfer reactions are very important to be considered specifically around and below the Coulomb barrier. The results were compared with the measured data and found in reasonable agreement.


INTRODUCTION
Nature displays one of its most impressive processes through the fusion of atomic nuclei.When the two nuclei merge with adequate kinetic energy they are capable of overcoming their electrostatic attraction to one another and producing a new nucleus with a charge and baryon number equal to the total of the original nuclei.The method of fusion depends on the amount of available kinetic energy, and The Coulomb barrier can be crossed or penetrated using quantum mechanical tunneling [1].In the creation of elements, the involvement of nuclei with a high neutron content is critical, whether it occurs naturally in astronomical events or experiments carried out on Earth.The production of heavy and superheavy nuclei using the neutron-richest projectiles and targets possible is favored by experimental and theoretical data [2][3].Neutron-rich nuclei fusion is thought to create heat that can cause an X-ray superburst in an accreting neutron star, according to a hypothesis [4].Where systems rich in neutrons are a rich material for research and exploration.In the investigation of the fusion of 58 Ni + 64 Ni at Q+2n = 3.9MeV, Beckerman et al first noticed the impact of Positive Q-value Neutron Transfer (PQNT) [5] channels on near-barrier fusion cross sections.They directly compared the fusion excitation functions of 58,64 Ni + 58,64 Ni in an experiment, with the symbol "+" notation representing the absorption of neutrons from the target nuclei [6].According to Broglia et al, the sub-barrier enhancement of fusion observed in 58 Ni + 64 Ni is due to a kinematic effect caused by the transfer of two neutrons during the fusion process, which reduces the neutron transfer cross-section [7] In some cases, the strength of the PQNT channels correlates strongly with the sub-barrier improvement of fusion [8].In comparison to 32 S+ 90 Zr, the subbarrier cross sections for 32 S+ 96 Zr are significantly greater.A coupled-channels calculation that considers the inelastic excitations can explain the sub-barrier improvement in 32 S+ 90 Zr only.Nonetheless, the unaccounted-for part of 32 S+ 96 Zr is thought to be connected to intermediate multi-neutron transfers with a positive-Q value.According to the similarities between 40 Ca+ 96 Zr and 32 S+ 96 Zr, couplings to positive-Q-value multi-neutron transfer channels may have improved sub-barrier fusion [9].It is noteworthy that the fusion cross sections for the 24 O+ 58 Ni and 40 Ca+ 96 Zr systems display different patterns in the vicinity of the barrier due to the contribution of the one-neutron transfer channel.While this channel has negligible impact on the 40 Ca+ 96 Zr reaction, it plays a significant role in the 24 O+ 58 Ni reaction.This discrepancy can be attributed to the distinct one-neutron transfer channel's Q-values, which are Q 1n = 5.29 MeV and Q 1n = 0.508 MeV for 24 O+ 58 Ni and 40 Ca+ 96 Zr, respectively [10].A density-constrained Time-Dependent Hartree-Fock (TDHF) [11] approach to fusion theory predicts increased fusion in the 24 O + 16 O system relative to the 16 O + 16 O system due to neutron transfer altering the potential and reducing the barrier.However, the fusion neutron-rich symmetric systems such as 24 O + 24 O is suppressed by a repulsive Pauli potential caused by the overlapping neutron-rich tails.[2] With the addition of one neutron to 18 O, the experimental fusion cross-section above the barrier rises by 37%, which is a remarkable outcome.The researcher's analysis showed that the rise in the fusion cross-section for 19 O is not the outcome of an odd-even effect, and the improvement in the fusion crosssection of 19 O is not a standard excitation.[4] Thought it was because of unpaired neutrons [12].When comparing neutronrich and non-neutron-rich systems, the N/Z ratio at the neck region is higher in the former.This increase in the N/Z ratio leads to a decrease in the Coulomb barrier, which enhances the fusion cross-sections in neutron-rich systems [13].Between interacting nuclei, neutron transfers can form a neck region of nuclear matter that encourages fusion.Once the nuclei are sufficiently close to one another to interact noticeably, or if positive Q-values for neutron transfers, neutron pick-up events take place [14].Broad distributions of experimental fusion cross-sections are produced by sequential neutron transfers.Neutron flow may result from the development of a neck between the projectile and target as a result of a collision with a finite Q-value [15].This could act as fusion's "doorway state".This intermediate state lowers the barrier and makes the fusion process easier at energies below the barrier, significantly increasing fusion cross-sections.An increase in sub-barrier fusion cross-sections has already been shown by experimental findings using (PQNT) [16][17].It has been proposed by Zagrebaev that the transfer of particles with negative Q values does not increase fusion at energies near or below the barrier [18].Based on a quantum diffusion approach, Sargsyan and co-workers suggested it if the deformation strength of the nuclei involved in neutron transfer does not change or decrease, neutron transfer channels have little impact on fusion dynamics.Consequently, in some instances, transfer channels may not lead to significant sub-barrier fusion enhancements despite (PQNT) [19].A universal fusion function approach was used by researchers to analyze the effects of (PQNT) on the fusion process in various systems.They found that significant deformation of the interacting nuclei following neutron transfer is crucial for strong sub-barrier fusion enhancement due to the (PQNT).In contrast, if the deformation of the nuclei is minimal or decreases after neutron transfer, these channels have little effect on the fusion cross-sections [20].
Majeed et al. сonducted an extensive study of the nuclear fusion process around and below the Coulomb barrier, the study included weakly bound nuclei 6 Li + 64 Ni, 11 B + 159 Tb, and 12 C + 9 Be using the semi-classical approach and full quantum mechanics, and they reached the results that the inclusion of the breakup channel is very important, To describe σ fus and D fus where results improve around and below the Coulomb barrier for light projectiles [21].To verify the results, different systems 6 Li+ 209 Bi, 7 Li+ 209 Bi , and 9 Be+ 208 Pb were studied.Similar results were reached in the previous research [22], by Fouad A. Majeed and Yousif A. Abdul-Hussien conducted a study on systems 17 F + 208 Pb and 15 C + 232 Th and found that the couplechannel effect between the elastic and continuous channel is very necessary for the calculations of σ fus and D fus where the results improved below and around the Coulomb barrier are very significant about the full quantum mechanics approach, [23].To confirm the results they reached, they did another study of the systems 6 He + 238 U and 8 He + 197 Au that improved the calculations around and below the Coulomb barrier [23].
The present study aims to investigate the effect of neutron or proton transfer and the impact of coupling between the elastic and breakup channels on the calculations of σ fus , D fus , and P fus , for the systems 28 Si + 90 Zr, 28 Si + 92 Zr, 28 Si + 94 Zr, 41 K + 28 Si, and 45 K + 28 Si.And compare the results with the related experimental data.

THE SEMICLASSICAL THEORY 2.1 No-Coupling or One-Channel Description
One-dimensional potential models have been utilized to evaluate the fusion C section -ross by employing the semiclassical theory.This model assumes that the degree of freedom of the colliding heavy ions can only be described by their relative motion [25,26].The Schrödinger equation is central to the semi-classical theory, which takes into account the energy and angular momentum of the system as well as the energy of the potential of the radial component of relative motion.Quantum tunneling is also considered in this theory.As a result, the semi-classical theory provides a comprehensive framework for understanding the behavior of quantum systems.cross-section of the fusion can be evaluated by implementing the semi-classical theory within the one-dimensional potential model [22].
In this context, the system's potential is denoted by (), while  represents the reduced mass of the system.To determine the wave-functions described in equation 1, one can utilize the time-dependent Schrodinger equations.To accomplish this, one can propose a particle trajectory based on Rutherford's usual trajectory.And include the real component of both the Coulomb potential and the centrifugal potential, which is expressed in [26].This methodology allows for the estimation of said wave-functions.
To consider profound absorption in the classically prohibited elastic channel scattering coupled-channel calculations, including the imaginary part of nuclear potential., which is complex, is essential [26].

𝑉𝑁(𝑟) = 𝑈𝑁(𝑟) − 𝑖𝑊(𝑟).
( Researchers have employed the wave expansion technique to investigate the considerable absorption resulting from the interference of (ℓ)waves, which originates from both the actual and imaginary components of the nuclear potential.Previous studies, as per the semi-classical theory, have demonstrated that fusion arises when two nuclei approach each other and pass through the potential barrier to enter the inner region.In this scenario, the WKB method can be utilized to determine the probability of penetration below the barrier.These findings have been documented in multiple sources, including references [25,27,28,29,30].
Eqn. 4 can be expressed as where the local wave-function number is ℓ (r), limits,  ℓ and  ℓ are as defined by places of turning to the classical trajectory.
Given that the fusion barrier can be adjusted using a parabolic function, the Hill-Wheeler formula [26] can be employed to depict the likelihood of penetration.
The fusion barrier cross-section can be calculated using equations [28,31] by employing the (WKB) approximation, where Ω and  () represent the curvature and height parameters of the barrier, respectively.The energy from the projectile bombing the target is indicated by E.
The radial component of the wave-function for the ℓ partial wave in the -channel is denoted by  (, ), while the potential imaginary part is represented by  ().

The Coupled Channels Formalism
To depict the collision, we're using the projectile-to-target separation vector.⃑ and the projectile's appropriate intrinsic degrees of freedom .To simplify, we do not consider the internal arrangement of the target.The Hamiltonian can be expressed as [32].
In this study, we ignore the nuclear coupling and focus solely on the interaction between the projectile and the target, which is represented by the term ( ⃗, ) in the intrinsic Hamiltonian of the projectile denoted as  ().We limit our theoretical comparison to the Coulomb dipole term.The equation [32] provides the eigenvectors of  ().
The energy of internal motion is denoted by ε β , and the Alder and Winther (AW) [33] approach involves two main steps.Firstly, classical mechanics is used to model the time growth of variable  ⃑, where the resulting trajectory is influenced by angular momentum ℓ and collision energy .In the original version of AW, a symmetrized Rutherford trajectory was utilized.However, in our case, the trajectory is determined by solving classical equations of motion with the potential ( ⃗) = ⟨ | ( ⃗, ) ⟩, where  represents the ground state of the projectile.This transforms the coupling interaction into a time-dependent interaction in the ξ-space, given by  ℓ (, ) ≡ ( ℓ ⃗(), ).The second step involves treating the dynamics in the intrinsic space as a problem in time-dependent quantum mechanics.This is achieved by expanding the wave function based on intrinsic eigenstates [33].
By plugging this spread into the Schrödinger equation for the wave function ψ(ξ, t), we arrive at the AW equations as described in reference [28].
To solve these equations, the initial conditions used were  (ℓ,  → −∞) =  .This signifies that the projectile was in its ground state before the collision ( → −∞).The resulting population of the channel after the collision corresponds to a specific angular momentum ℓ :  ℓ =  (ℓ,  → +∞) , the cross-section of the angle-integrated [33] To apply this technique to fusion reactions, we begin by utilizing quantum mechanics to compute the fusion cross section in a coupled-channel scenario.To simplify the process, we assume that all channels are bound and have no spins.
The Effect of the Proton and Neutron as Probe for the Nuclear Fusion Reactions...

EEJP. 3 (2023)
The total fusion cross-section can be obtained by adding up the contributions from each channel.By conducting partialwave expansions, we arrive at the following equation as shown in reference [34].
The optical potentials of the imaginary part connected to fusion in the channel ( ℓ  ,  ) above have an absolute value of ( ) and the radial wave function for the ℓth partial wave is () in the equation above.
One possible way to estimate the likelihood of fusion occurring is by using the following approximation: Let  ℓ denote the transmission factor through the barrier, and  (ℓ,  ) be the probability of finding the system in the channel  at the point of closest approach ( ).[35][36]

Distribution of Fusion Barrier
Accurately determining the distribution of the fusion barrier parameter  is crucial, as it is highly sensitive and has been defined in previous research [29,31].The function denoted as () represents the distribution of fusion barriers and is defined as follows: Considerable progress has been made in comprehending the fusion reaction by establishing the reaction fusion barrier distribution through experimentation.To determine the uncertainties in numerical fusion barrier calculations, it is possible to refer to the reaction [37,38].
In this case, the cross-sectional data were measured at various excitation energy points, with an interval of ∆.To determine the statistical error, the relation [29] was utilized.
The uncertainty in the product of () for each collision energy, denoted by F(), is given as [29].

RESULTS AND DISCUSSION
This section presents the theoretical calculations of the fusion reaction σ fus , the fusion barrier distribution D fus and the fusion probability P fus .These calculations were obtained using the semi-classical theory with a Continuum Discretized Coupled Channel (CDCC) [39] approach to study the effect of calculating the coupling between elastic channels and breakup channels on fusion processes.The calculations were performed using the Sequential Complete Fusion SCF [40] code for semi-classical comparisons, while the quantum mechanical calculations were performed using the CC code for the systems, 28 Si + 90 Zr, 28 Si + 92 Zr, 28 Si + 94 Zr, 41 K + 28 Si, and 45 K + 28 Si.Table 1 presents the parameters for the Akyüz-Winther potential, with a coulomb barrier.1.The three drawings below depict the theoretical and experimental ,  and  for the system, obtained through a combination of quantum mechanical calculations and semi-classical methods.The data used in this study are taken from Ref. [41].The blue curves correspond to the quantum mechanical calculations with coupling and no-coupling, respectively.The blue curve with the spaces shows the results obtained with no-coupling.Similarly, the red curves, correspond to the semi-classical calculations with and no-coupling, respectively.The dashed red curve shows the results obtained no-coupling.The system's data regarding the position of the Coulomb sub-barrier V b are sourced from the black arrow.The semi-classical calculations, no-coupling and coupling have a great concurrence with experimental data after exceeding V b , and for below V b, there is no concurrence with experimental data.For no-coupling and coupling quantum mechanical calculation, after exceeding V b there is a special concurrence. drawing (C), alongside experimental data [41] for the system 28 Si+ 90 Zr.
4.2 ( 28 Si+ 92 Zr) reaction Figure 2. The three drawings below depict the theoretical and experimental ,  and  for the system, obtained through a combination of quantum mechanical calculations and semi-classical methods.The data used in this study are taken from Ref. [42].The blue curves correspond to the quantum mechanical calculations with coupling and no-coupling, respectively.The blue curve with the spaces shows the results obtained with no-coupling.Similarly, the red curves, correspond to the semi-classical calculations with and no-coupling, respectively.The dashed red curve shows the results obtained no-coupling.The system's data regarding the position of the Coulomb sub-barrier V b are sourced from the black arrow.The semi-classical calculations, no-coupling and coupling have a special concurrence with experimental data after exceeding V b and below V b .For no-coupling and coupling quantum mechanical calculation, after exceeding V b there is a special concurrence. drawing (C), alongside experimental data [42] for the system 28 Si+ 92 Zr.

( 28
Si+ 94 Zr) reaction Figure 3.The three drawings below depict the theoretical and experimental ,  and  for the system, obtained through a combination of quantum mechanical calculations and semi-classical methods.The data used in this study are taken from Ref. [41].The blue curves correspond to the quantum mechanical calculations with coupling and no-coupling, respectively.The blue curve with the spaces shows the results obtained with no-coupling.Similarly, the red curves, correspond to the semi-classical calculations with and no-coupling, respectively.The dashed red curve shows the  4.4.( 41 K+ 28 Si) reaction Figure 4.The three drawings below depict the theoretical and experimental ,  and  for the system, obtained through a combination of quantum mechanical calculations and semi-classical methods.The data used in this study are taken from Ref. [43].The blue curves correspond to the quantum mechanical calculations with coupling and no-coupling, respectively.The blue curve with the spaces shows the results obtained with no-coupling.Similarly, the red curves, correspond to the semi-classical calculations with and no-coupling, respectively.The dashed red curve shows the results obtained no-coupling.The system's data regarding the position of the Coulomb sub-barrier V b are sourced from the black arrow.The semi-classical calculations, no-coupling have a special concurrence with experimental data after exceeding V b , and for below V b there is no concurrence, and for coupling, there is a great concurrence.For quantum mechanical calculation, there is a great concurrence after exceeding V b . drawing (C), alongside experimental data [43] for the system 41 K+ 28 Si.

4.5.
( 45 K+ 28 Si) reaction Figure 5.The three drawings below depict the theoretical and experimental ,  and  for the system, obtained through a combination of quantum mechanical calculations and semi-classical methods.The data used in this study are taken from Ref. [43].The blue curves correspond to the quantum mechanical calculations with coupling and no-coupling, respectively.The blue curve with the spaces shows the results obtained with no-coupling.Similarly, the red curves, correspond to the semi-classical calculations with and no-coupling, respectively.The dashed red curve shows the results obtained no-coupling.The system's data regarding the position of the Coulomb sub-barrier V b are sourced from the black arrow.The semi-classical calculations, no-coupling and coupling have a good concurrence with experimental data after exceeding V b .For no-coupling and coupling quantum mechanical calculation, below V b there is a great concurrence.[43] for the system 45 K+ 28 Si.

CONCLUSION
The effect of channel coupling between elastic channels and breakup channels on calculating fusion cross-section (σ fus ), fusion barrier distribution (D fus ), and fusion probability (P fus ), on systems 28 Si + 90 Zr, 28 Si + 92 Zr, 28 Si + 94 Zr, 41 K + 28 Si, and 45 K + 28 Si were investigated in the present study and these systems involved one neutron or one proton transfer reactions.Our theoretical calculations showed that the results improved significantly below the Coulomb barrier, especially the semi-classical calculations.The reason for the improvement below the Coulomb barrier can be attributed to the fact that coupling effects become more significant at lower energies, leading to a more complex interaction between the elastic channels and the breakup channels.Introducing coupling conditions into the calculations takes into account the exchange of flow between these channels, which results in better agreement with the experimental data.On the other hand, the slight overestimation of the results above the Coulomb barrier can be attributed to the fact that the fusion process becomes more controlled by the elastic channel at higher energies, and the coupling effects become less significant.The reason for the overestimation of the cross-section may be the fact that our calculations may not have fully accounted for the effect of the Coulomb barrier on the merger process.

Figure 1 .
Figure 1.Displays the results of quantum mechanical and semi-classical calculations for the  drawing (A),  drawing (B), and drawing (C), alongside experimental data[41] for the system 28 Si+ 90 Zr.

Figure 2 .
Figure 2. Displays the results of quantum mechanical and semi-classical calculations for the  drawing (A),  drawing (B), and drawing (C), alongside experimental data[42] for the system 28 Si+ 92 Zr.
no-coupling.The system's data regarding the position of the Coulomb sub-barrier V b are sourced from the black arrow.The semi-classical calculations, no-coupling and coupling have a good concurrence with experimental data after exceeding V b , and for below V b , there is a special concurrence.For no-coupling and coupling quantum mechanical calculation, there is no concurrence.

Figure 3 .
Figure 3. Displays the results of quantum mechanical and semi-classical calculations for the  drawing (A),  drawing (B), and drawing (C), alongside experimental data[41] for the system 28 Si+ 94 Zr.

Figure 4 .
Figure 4. Displays the results of quantum mechanical and semi-classical calculations for the  drawing (A),  drawing (B), and drawing (C), alongside experimental data[43] for the system 41 K+28 Si.

Figure 5 .
Figure 5. Displays the results of quantum mechanical and semi-classical calculations for the  drawing (A),  drawing (B), and drawing (C), alongside experimental data[43] for the system 45 K+28 Si.

Table 1 .
The Coulomb barrier height Vb and The Akyüz-Winther potential parameters 4.1.