The coupled channel method

The CCFULL computer code was applied to calculate the fusion cross-sections [3, 11], the incoming wave boundary condition (IWBC) was imposed in the calculations, and the absorption radius was considered to be at the minimum of the potential pocket inside the Coulomb barrier. With the IWBC, the coupled-channel equations can be given by the following [3]: $\begin{array}{l}[-\frac{{\hslash }^2}{2\mu }\frac{{\text{d}}^2}{\text{d}{R}^2}+\frac{J(J+1){\hslash }^2}{2\mu R^2}+V(R)+{ϵ}_n-E]u_n(R)\\ +{\displaystyle \sum _m}{V}_{nm}(R)u_m(R)=\mathrm{0,}\end{array}$ (1) where E is the incident energy in the center-of-mass frame, and ϵn and un are the excitation energy and radial wave function of the nth channel, respectively. The total potential V(R) between two colliding nuclei consists of the nuclear and Coulomb interactions, that is, V(R)=V_N(R)+V_C(R). The nuclear V_N(R) and Coulomb V_C(R) interactions used in the calculations were both obtained using the double-folding procedure.

The symbol Vnm(R) in Eq.(1) denotes the matrix of the coupling Hamiltonian, which includes both Coulomb and nuclear components. The Coulomb coupling matrix elements $V_{nm}^{\text{C}}$ were calculated using linear coupling approximation [3, 11, 1]. The nuclear coupling Hamiltonian was generated by introducing a dynamical operator ${\widehat{O}}_{\lambda }$ in the calculations, which is given by ${\tilde{V}}_{\text{N}}(R\mathrm{,}{\widehat{O}}_{\lambda })=V_{\text{N}}(R-{\widehat{O}}_{\lambda })$ [3, 11]. For vibrational coupling, the operator ${\widehat{O}}_{\lambda }$ is given by ${\widehat{O}}_{\lambda }=({\beta }^{*}/\sqrt{4\pi })R_i({\alpha }_{\lambda 0}^{†}+{\alpha }_{\lambda 0})$ [3, 11, 1], where ${\alpha }_{\lambda 0}^{†}$ and α_{λ 0} are the creation and annihilation operators of the phonons, respectively, the eigenvalues λ and eigenvectors $|\alpha �\#x27E9;$ of the operator $\widehat{O}$ satisfy ${\widehat{O}}_{\lambda }|\alpha �\#x27E9;={\lambda }_{\alpha }|\alpha �\#x27E9;$, Ri is the radius of the projectile or target, and β^* denotes the corresponding deformation parameter. The nuclear coupling matrix elements were then evaluated using [3] $\begin{array}{c}{V}_{nm}^{\text{N}}=\langle n|{\tilde{V}}_{\text{N}}(R\mathrm{,}{\widehat{O}}_{\lambda })|m\rangle -V_{\text{N}}(R){\delta }_{n\mathrm{,}m}\\ ={\displaystyle \sum _{\alpha }}\langle n|\alpha \rangle \langle \alpha |m\rangle {\tilde{V}}_{\text{N}}(R\mathrm{,}{\lambda }_{\alpha })-V_{\text{N}}(R){\delta }_{n\mathrm{,}m}\mathrm{.}\end{array}$ (2) The nuclear coupling potential ${\tilde{V}}_{\text{N}}(R\mathrm{,}{\lambda }_{\alpha })=V_{\text{N}}(R-{\lambda }_{\alpha })$ is taken up to the second-order of λα [3, 11, 1] ${\tilde{V}}_{\text{N}}(R\mathrm{,}{\lambda }_{\alpha })=V_{\text{N}}(R)-\frac{\text{d}{V}_{\text{N}}(R)}{\text{d}R}{\lambda }_{\alpha }+\frac{1}{2}\frac{{\text{d}}^2{V}_{\text{N}}(R)}{\text{d}{R}^2}{\lambda }_{\alpha }^2\mathrm{,}$ (3) where the first term V_N(R) is the nuclear potential in the absence of coupling, and the second and third terms are the nuclear coupling form factors, which are closely associated with the nuclear potential.

The penetrability PJ can be obtained by solving the CC equations, and the total fusion cross-section σ_fus is then obtained by summing the partial fusion cross-section [3]: ${\sigma }_{\text{fus}}(E)=\frac{\pi }{k^2}{\displaystyle \sum _J}(2J+1)P_J(E)\mathrm{,}$ (4) where $k=\sqrt{2\mu E/{\hslash }^2}$ is the wave number associated with energy E.

The double folding potential

The explicit form of the potential between the projectile and target nuclei is presented herein. The double-folding model was used to calculate the nucleus-nucleus potential. This model has been widely used to analyze the elastic and inelastic scattering of heavy ions [46, 47]. However, it is difficult to explain the fusion process in a strong-density overlap region [48]. When the projectile and target nuclei start touching one another, the Pauli blocking effects become increasingly important owing to the density overlap. Considering the Pauli blocking potential resulting from the antisymmetrization, a modified double-folding potential was employed to analyze the heavy-ion fusion processes as follows [21]: $\begin{array}{l}{V}_{\text{N}}(R)=V_{\text{D}}(R)+V_{\text{P}}(R)\\ ={\displaystyle \int }\text{d}{r}_1\text{d}{r}_2{\rho }_{\text{p}}(r_1){\rho }_{\text{t}}(r_2)g(|s_1|)\\ +{\displaystyle \int }\text{d}{r}_1{\rho }_{\text{p}}(r_1)v(|s_2|)\mathrm{,}\end{array}$ (5) $g(|s_1|)=9846\frac{\exp (-4s_1)}{4s_1}-3139\frac{\exp (-2.5s_1)}{2.5s_1}\mathrm{,}$ (6) $v(|s_2|)=437.05{\rho }_{\text{t}}(s_2)+983.89{\rho }_{\text{t}}^2(s_2)\mathrm{.}$ (7) where $g(|s_1=R-r_1+r_2|)$ denotes the direct interaction between a nucleon in the target nucleus and a nucleon in the projectile nuclei, and $v(|s_2=R-r_1|)$ denotes the Pauli blocking interaction of a single nucleon in the projectile nuclei from the target density. ρ_p and ρ_t denote the density distributions of the projectile and target nuclei, respectively. This nuclear potential, including the Pauli blocking effect, has been successful in applying the fusion processes of 95 systems and has significantly improved the explanation of fusion cross sections at deep sub-barrier energies [21]. The Coulomb potential employed in the calculations was the double-folding integral of the proton-proton Coulomb interaction: $V_{\text{C}}(R)={\displaystyle \int }\text{d}{r}_1\text{d}{r}_2\frac{e^2}{\left|s_1\right|}{\rho }_{\text{pp}}(r_1){\rho }_{\text{tp}}(r_2)\mathrm{,}$ (8) where ρ_pp and ρ_tp are the proton density distributions of the projectile and target nuclei, respectively, and e is the elementary charge.

The density distributions of the projectile and target nuclei are usually parameterized with Fermi-Dirac distribution functions [49]: ${\rho }_i(r)=\frac{{\rho }_{0i}}{1+\exp (\frac{r-C_i}{a_i})}\mathrm{,}$ (9) where ρi, Ci, and ai indicate the density, half-density radius, and diffuseness of the neutrons (i=n) and protons (i=p), respectively. ρ_0i is obtained by integrating the matter density distribution that is equivalent to the neutron or proton numbers. In the sudden or frozen density approximation (FDA), the densities of the two colliding nuclei are assumed to be frozen for simplicity, that is, the parameters of ρ_i0, Ci, and ai in Eq. (9) are fixed during the fusion process.

The non-frozen density approximation

In the adiabatic model, the fusion reaction is assumed to occur slowly, thus the density distribution has sufficient time to adjust to the optimized distribution [3]. To describe the adiabatic picture, several different methods were employed to determine the transformation process of the density distribution and obtain the adiabatic potential after the overlap of the two colliding nuclei [38, 43, 50, 51]. In [43], Reichstein and Malik introduced a special nonfrozen density approximation (NFDA), the ¹⁶O + ¹⁶O fusion reaction. A similar approach was recently successfully employed to describe the elastic scattering and fusion data for the ¹²C + ¹²C reaction [44, 45]. This approach introduces the distance dependence R to the density parameters C and a as follows: $C(R)=\{\begin{array}{c}C\mathrm{,}R>R_{\text{c}}\\ C_{\text{cn}}\exp [\ln (\frac{C}{C_{\text{cn}}})\cdot \frac{R^2}{R_{\text{c}}^2}]\mathrm{,}R\le R_{\text{c}}\end{array}$ (10) and the same hypothesis for the diffuseness parameter a. Here, C_cn is the half-density radius of the compound nucleus and R_c is the distance between the projectile and target, where the density of the overlap region is the central density of the compound nuclei. At the beginning of the density overlap or before approaching the distance R_c, the density distribution of the two colliding nuclei can be considered frozen. In the strong density overlap region, namely R≤R_c, there is a density transformation from two colliding nuclei to a compound nucleus.

Considering this non-frozen process, the standard CC framework is necessary to modify it by introducing a damping factor that describes the physical process for transitioning from the sudden to adiabatic approximations [37, 38]. To avoid the double counting of the CC effects, we introduced the following damping factor to simulate the decrease in the excitation strengths of the colliding nuclei, such as the vibrational states: [38] $\Phi (R\mathrm{,}{\lambda }_{\alpha })=\{\begin{array}{cc}\exp [-{(R-R_{\text{d}}-{\lambda }_{\alpha })}^2/2a_{\text{d}}^2]\mathrm{,}& R\le R_{\text{d}}+{\lambda }_{\alpha }\mathrm{,}\\ \mathrm{1,}& R>R_{\text{d}}+{\lambda }_{\alpha }\mathrm{,}\end{array}$ (11) where R_d and a_d are parameters indicating the damping radius and diffuseness, respectively. Instead of Eq. (3) in the CC model, the nuclear coupling potential was employed [37, 38]: $\begin{array}{c}{\tilde{V}}_{\text{N}}(R\mathrm{,}{\lambda }_{\alpha })=V_{\text{N}}(R)-[\frac{\text{d}{V}_{\text{N}}(R)}{\text{d}R}{\lambda }_{\alpha }\\ +\frac{1}{2}\frac{{\text{d}}^2{V}_{\text{N}}(R)}{\text{d}{R}^2}{\lambda }_{\alpha }^2]\Phi (R\mathrm{,}{\lambda }_{\alpha })\mathrm{.}\end{array}$ (12) The CC equation performed sufficiently at large distances. With a decrease in distance, the non-frozen process gradually plays a leading role, and the CC model is close to the one-dimensional potential model.

the medium-heavy mass system ⁶⁴Ni + ⁶⁴Ni

First, the density distribution of the FDA and NFDA after two colliding nuclei overlap is presented in Fig. 1. Based on the FDA, the densities of the two colliding nuclei are independent of the distance R and are fixed. However, the NFDA describes a density transformation from the colliding nuclei to the compound nucleus. Before distance R_c, the two colliding nuclei maintain their individuality. After reaching R_c, the half-density radii C of the projectile and target begin to increase, and the nucleons in the center region of the projectile and target gradually diffuse to the edge, that is, the compact nuclei dissolve. According to the normalization condition, the central densities of the projectile and target nuclei decreased, and their surface densities increased. For the ⁶⁴Ni + ⁶⁴Ni system, before the distance reached R_c = 8.6 fm, the overlap density distributions obtained from the FDA and NFDA were the same, as indicated by the dashed lines in Fig. 1. At R = 6.5 fm, the two results indicate that the fusion processes at small distances are different and the density distribution of the NFDA is looser. Note, at the initial stage of density rearrangement (8.0 fm R < 8.6 fm), the density overlap region is mainly contributed by the colliding nuclei in the diffuseness region. Therefore, the density at Z = 0 fm obtained from the NFDA was slightly higher than that obtained from the FDA; as shown in Fig. 1, at a distance R = 8.3 fm, the maximum density of NFDA was 0.1900 fm^-3 and only 0.1880 fm^-3 for FDA. This slight difference that results from the surface density rearrangement has a significant influence on the nucleus-nucleus potential presented below.

Fig. 1Full-screen View

(Color online) Comparison of the overlap density distribution obtained from the FDA and NFDA at the varying distance R for the ⁶⁴Ni + ⁶⁴Ni fusion system. The distance R_c used in Eq. (10) is 8.6 fm, where the density of the overlap region is equal to the central density of the compound nuclei ¹²⁸Ba.

Here, we provide a detailed comparison of the potentials demonstrated in Fig. 2 for the ⁶⁴Ni + ⁶⁴Ni fusion system. Before reaching a distance of 8.6 fm, the potentials obtained from the FDA and NFDA were the same, and the difference became increasingly apparent as the distance R decreased. At the initial stage of the density rearrangement, the surface density rearrangement indicated above leads to the potential of the NFDA being more repulsive for the Pauli blocking term and more attractive for the direct term compared to the FDA results, as shown in Fig. 2(a) and (b). However, as the nuclei become distant, the Pauli blocking potential and the direct potential obtained from the NFDA have weaker effects, corresponding to a weaker density overlap. As the sum of V_P and V_D, the nuclear potential V_N shown in Fig. 2(c) resulted from the NFDA becoming more attractive at a small radii compared to the result of the FDA. As shown in Fig. 2(d), the long-range Coulomb potential, which is not affected by this surface density rearrangement, is always less repulsive by considering the NFDA.

Fig. 2Full-screen View

(Color online) Comparison of the potential obtained from FDA (solid line) and NFDA (dashed line) for the ⁶⁴Ni + ⁶⁴Ni fusion system. (a), (b), and (c) present the Pauli blocking term V_P, direct term V_D, and the nuclear interaction V_N in Eq. (5), respectively. (d) presents the Coulomb interaction V_C in Eq. (8). The shadow zone denotes the area of distance R > 8.6 fm.

The total potentials and fusion cross-sections are shown in Fig. 3. The commonly used potentials in heavy-ion fusion reactions, such as the Akyüz-Winther (AW) potential [39], are also presented in Fig. 3(a) for comparison. Two double folding potentials, namely FDA and NFDA, exhibit similar behavior to the AW potential outside the Coulomb barrier and generate shallow pockets inside the Coulomb barrier owing to the Pauli blocking effect. The two potentials obtained from the FDA and NFDA are distinguished after two colliding nuclei approach the distance R = 8.6 fm, where the density distributions of the colliding nuclei begin to change. At the initial stage of the density rearrangement, the stronger density overlap of NFDA resulted in a slightly shallower pocket than that of FDA. With a decrease in the distance R, the two potentials demonstrated apparently different energy dependencies, and the NFDA became more attractive. However, this difference between the two potentials inside the pocket does not contribute to the calculated fusion cross sections owing to the IWBC adopted in the heavy-ion fusion model.

Fig. 3Full-screen View

(Color online) (a) Comparison of the total potentials obtained from AW (dotted line), FDA (dashed line), and NFDA (solid line) for the ⁶⁴Ni + ⁶⁴Ni fusion system. (b) The experimental fusion cross sections for the ⁶⁴Ni + ⁶⁴Ni [54] fusion system compared to the theoretical calculation results. The dotted and dashed lines denote the results of the AW and FDA potentials. The results calculated by the NFDA potential with and without the damping factor are displayed by the dash-dotted and solid lines, respectively. The insert presents a comparison of the S factors between the experimental data and calculated results.

In Fig. 3(b), the calculated fusion cross sections based on the CC model are compared with the experimental data [54]. The experimental fusion cross-sections near the Coulomb barrier energies are underestimated by the no-coupling results (thin solid line) and are sufficiently described by considering the CC effect. As the collision energies decrease to deep sub-barrier energies, the experimental data exhibit a strong suppression compared to the AW results (dotted line). This phenomenon is referred to as fusion hindrance and mainly results from the density overlap in the fusion process. Although the fusion cross sections calculated by FDA, which include the Pauli blocking potential, present a significant improvement, it remains difficult for the astrophysical S factor obtained from the FDA to represent the experimental maximum at deep sub-barrier energies, as shown in the insert of Fig. 3(b). Considering the similar behavior of the FDA and NFDA potential outside the pocket, there was no apparent difference between the fusion cross-sections of the FDA and NFDA. However, considering the damping effect of the CC framework resulting from the density arrangement, the calculated fusion cross-sections presented a steep falloff at deep sub-barrier energies, sufficiently presenting the experimental data. Importantly, the S factor obtained from the NFDA with the damping factor ceases to increase and appears to be maximum at deep sub-barrier energies.

the medium-light mass system ²⁴Mg + ³⁰Si

Figure 4 presents the obtained total potential and fusion cross sections for the medium-light mass system ²⁴Mg + ³⁰Si. Similar to the ⁶⁴Ni + ⁶⁴Ni fusion system, considering the non-frozen process, the potential from colliding nuclei to the compound nucleus, namely the NFDA, also presented a slightly shallower pocket compared to the result of FDA shown in Fig. 4(a). The fusion cross-sections obtained from the FDA and NFDA with or without the damping factor sufficiently explain the fusion hindrance to a certain extent, as shown in Fig. 4(b). By introducing the damping factor, the fusion cross-sections of the NFDA presented a stronger energy dependence at deep sub-barrier energies and then rapidly converged to the results indicating no coupling. The S factors calculated by the FDA and NFDA, with or without the damping factor, were all in good agreement with the experimental data. Although the growth of the slope for the S factor slowed with decreasing energies, the maximum of the S factor calculated by NFDA with the damping factor is not visible in the insert shown in Fig. 4(b).

Fig. 4Full-screen View

(Color online) Same as Fig. 3, but for the fusion system ²⁴Mg + ³⁰Si. The solid points indicate the experimental data obtained from Refs. [52, 55]

In comparison with the medium-light mass system ²⁴Mg + ³⁰Si, the density rearrangement of the ⁶⁴Ni + ⁶⁴Ni system in the overlap region has a significant impact on the fusion and presents a relatively apparent damping effect of the CC strength. The S factor calculated with the damping factor ceases to increase and exhibits a maximum at deep sub-barrier energies in the ⁶⁴Ni + ⁶⁴Ni fusion system. However, in the relatively light mass system ²⁴Mg + ³⁰Si, the growth of the S factor at lower energies was slowed by considering the non-frozen condition with a relatively weak damping effect. This difference in the behavior of the S factor may provide a valuable reference for the currently controversial fusion process of ¹²C + ¹²C at low energies, which strongly affects the estimation of astrophysical reaction rates.