Non-frozen process of heavy-ion fusion reactions at deep sub-barrier energies

Non-frozen process of heavy-ion fusion reactions at deep sub-barrier energies 1 corresp first-author chengkaixuan@htu.edu.cn chengkaixuan@htu.edu.cn chengkaixuan@htu.edu.cn https://orcid.org/0000-0003-0067-3726 1 https://orcid.org/0000-0001-7169-4044 1 1 1 corresp machunwang@126.com machunwang@126.com machunwang@126.com School of Physics, Henan Normal University, Xinxiang 453007, China School of Physics, Henan Normal University, Xinxiang 453007, China The hindrance in heavy-ion fusion reactions at deep sub-barrier energies is investigated using the double folding model with a hybrid method between the frozen and adiabatic density approximations. In this method, the density distributions of the projectile and the target depend closely on the distance between them. As the distance decreased, the half-density radii of the colliding nuclei gradually increased to the half-density radius of the compound nucleus. The total potential based on this non-frozen approximation generates a slightly shallower pocket and becomes more attractive inside the pocket compared to that obtained from the frozen approximation. A damping factor was used to simulate the decline of the coupled channel effects owing to the density rearrangement of the two colliding nuclei. The calculated fusion cross-sections and astrophysical S factors at the deep sub-barrier energies are both in good agreement with the experimental data for the medium-heavy 64 Ni + 64 Ni and medium-light 24 Mg + 30 Si mass systems. In addition, it was concluded that the apparent maximum of the S factors most likely appear in fusion systems with strong coupling effects. Adiabatic approximation Double folding model Fusion hindrance Introduction 1 Heavy-ion fusion reactions are important for understanding the fundamental properties of quantum tunneling in complex many-body systems. As fusion occurs, the relative motion of the two colliding nuclei must overcome a Coulomb barrier, which is composed of the short-range attractive nuclear potential and the long-range repulsive Coulomb potential. When the collision energy is near or above the Coulomb barrier, a simple one-dimensional potential model sufficiently describes the fusion process. However, at energies below the Coulomb barrier, the coupling effects between the relative motion of the colliding nuclei and the nuclear intrinsic degrees of freedom, such as the vibrations and rotations of the target or projectile, play an important role in quantum tunneling [ 1 1 - 3 3 1-3 ]. A significant increase in fusion cros s-sections at sub-barrier energies was observed compared to the results of a simple one-dimensional potential model [ 4 4 - 8 8 4-8 ]. Considering these coupling effects, the coupled-channel (CC) model has been successfully used in several fusion reaction calculations [ 9 9 - 12 12 9-12 ]. Recently, at collision energies significantly below the Coulomb barrier, an unexpected steep falloff feature of the experimental fusion cross-section, in contrast to the theoretical prediction, presents a new challenge to the current fusion theory [ 13 13 - 15 15 13-15 ]. This steep falloff feature in the experimental data, also called "fusion hindrance", is difficult to explain by the CC model and has gained widespread attention once again in heavy-ion fusion reactions [ 16 16 - 23 23 16-23 ], especially astrophysical fusion reactions such as 12 C + 12 C [ 24 24 - 29 29 24-29 ]. The key to understanding this hindrance phenomenon is dealing with the density of the composite system after two colliding contacts. Thus far, two different assumptions have been proposed to describe the transformation from a projectile and target to a compound nucleus; namely, the sudden and adiabatic approaches. Based on the sudden or frozen density approximation, it was assumed that the densities of the two colliding nuclei were frozen. Therefore, there should be a strong repulsive core in the nuclear potential of heavy-ion fusion systems in strong density overlap regions, resulting from the incompressibility properties of nuclear matter [ 30 30 - 32 32 30-32 ] or the Pauli exclusion principle of many-body quantum systems comprising fermions [ 33 33 - 36 36 33-36 ]. On the other hand, in the adiabatic approach, fusion is considered to occur slowly, thus allowing the density of the composite system to have sufficient time to rearrange. In [ 37 37 , 38 38 ], Ichikawa et al. constructed an adiabatic one-body potential to describe the neck formation between two colliding nuclei in the overlap region, and a damping factor was introduced in the standard CC model to describe the physical process for the gradual transition from the sudden to adiabatic approximations. Because the assumptions employed by these two methods are completely contradictory, the depths of the potential pockets inside the Coulomb barrier significantly differ [ 38 38 , 3 3 ]. In addition, as a common feature, the potentials obtained from these two methods are both wider than the commonly used Akyüz-Winther (AW) potential [ 39 39 ], which is successful in describing the fusion reactions above and at sub-barrier energies; however it is difficult to explain the fusion hindrance at deep sub-barrier energies [ 16 16 ]. As a reverse quantum tunneling process with fusion, radioactive α decay in heavy nuclei was successfully conducted by microscopic calculations, revealing significant details regarding the internal structure of nuclei [ 40 40 - 42 42 40-42 ]. For example, in [ 40 40 ], Röpke et al. concluded that owing to strong Pauli blocking effects, the α particle is highly sensitive to the surrounding matter. When approaching the nucleus 208 Pb, the α particle dissolves in the density overlap region and its four nucleons mix with the surrounding matter, which indicates that the α -particle size increases and the density distribution changes with a decrease in the distance between the center of mass of the α particle and the 208 Pb nucleus. Furthermore, to remove the sudden approximation in heavy-ion fusion reactions, Reichstein and Malik employed a special non-frozen approach to calculate the 16 O + 16 O potential by introducing a distance dependence to the density parameters, demonstrating a transformation from the density distributions of the reactants to that of a compound nucleus [ 43 43 ]. This method was also recently employed to study the elastic scattering and fusion reactions of the 12 C + 12 C system [ 44 44 , 45 45 ]. In this study, instead of the frozen density approximation, the non-frozen approximation was employed by the double folding model to analyze the fusion process in a strong density overlap region. Considering 64 Ni + 64 Ni and 24 Mg + 30 Si fusion reactions as examples, the potentials based on the non-frozen and frozen density approximations are compared in detail. The fusion cross-sections, particularly the hindrance at deep sub-barrier energies, and the astrophysical S factor were both investigated by considering this non-frozen approach. In Sect. 2, we provide an explicit form of the double folding potential and briefly describe the non-frozen method. The differences between the frozen potential and non-frozen potential, as well as the fusion cross-sections, are presented in Sect. 3. A summary is presented in Sect. 4. Theoretical framework 1 The coupled channel method 2 The CCFULL computer code was applied to calculate the fusion cross-sections [ 3 3 , 11 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 3 ]: <math display="block" id="M1"><mtable columnalign="left"><mtr><mtd><mo stretchy="false">[</mo><mo>−</mo><mfrac><mrow><msup><mi>ℏ</mi><mn>2</mn></msup></mrow><mrow><mn>2</mn><mi>μ</mi></mrow></mfrac><mfrac><mrow><msup><mtext>d</mtext><mn>2</mn></msup></mrow><mrow><mtext>d</mtext><msup><mi>R</mi><mn>2</mn></msup></mrow></mfrac><mo>+</mo><mfrac><mrow><mi>J</mi><mo stretchy="false">(</mo><mi>J</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><msup><mi>ℏ</mi><mn>2</mn></msup></mrow><mrow><mn>2</mn><mi>μ</mi><msup><mi>R</mi><mn>2</mn></msup></mrow></mfrac><mo>+</mo><mi>V</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mi>ϵ</mi><mi>n</mi></msub><mo>−</mo><mi>E</mi><mo stretchy="false">]</mo><msub><mi>u</mi><mi>n</mi></msub><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mtext>                </mtext><mo>+</mo><mstyle displaystyle="true"><munder><mo>∑</mo><mi>m</mi></munder></mstyle><msub><mi>V</mi><mrow><mi>n</mi><mi>m</mi></mrow></msub><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo><msub><mi>u</mi><mi>m</mi></msub><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0,</mn></mtd></mtr></mtable></math> where E is the incident energy in the center-of-mass frame, and ϵ n and u n are the excitation energy and radial wave function of the n th 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 V nm ( R ) in Eq.(1) denotes the matrix of the coupling Hamiltonian, which includes both Coulomb and nuclear components. The Coulomb coupling matrix elements <math id="M2"><mrow><msubsup><mi>V</mi><mrow><mi>n</mi><mi>m</mi></mrow><mtext>C</mtext></msubsup></mrow></math> were calculated using linear coupling approximation [ 3 3 , 11 11 , 1 1 ]. The nuclear coupling Hamiltonian was generated by introducing a dynamical operator <math id="M3"><mrow><msub><mover accent="true"><mi>O</mi><mo>^</mo></mover><mi>λ</mi></msub></mrow></math> in the calculations, which is given by <math id="M4"><mrow><msub><mrow><mover accent="true"><mi>V</mi><mo stretchy="true">˜</mo></mover></mrow><mtext>N</mtext></msub><mo stretchy="false">(</mo><mi>R</mi><mn>,</mn><msub><mover accent="true"><mi>O</mi><mo>^</mo></mover><mi>λ</mi></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mi>V</mi><mtext>N</mtext></msub><mo stretchy="false">(</mo><mi>R</mi><mo>−</mo><msub><mover accent="true"><mi>O</mi><mo>^</mo></mover><mi>λ</mi></msub><mo stretchy="false">)</mo></mrow></math> [ 3 3 , 11 11 ]. For vibrational coupling, the operator <math id="M5"><mrow><msub><mover accent="true"><mi>O</mi><mo>^</mo></mover><mi>λ</mi></msub></mrow></math> is given by <math id="M6"><mrow><msub><mover accent="true"><mi>O</mi><mo>^</mo></mover><mi>λ</mi></msub><mo>=</mo><mo stretchy="false">(</mo><msup><mi>β</mi><mo>*</mo></msup><mo>/</mo><msqrt><mrow><mn>4</mn><mi>π</mi></mrow></msqrt><mo stretchy="false">)</mo><msub><mi>R</mi><mi>i</mi></msub><mo stretchy="false">(</mo><msubsup><mi>α</mi><mrow><mi>λ</mi><mn>0</mn></mrow><mo>†</mo></msubsup><mo>+</mo><msub><mi>α</mi><mrow><mi>λ</mi><mn>0</mn></mrow></msub><mo stretchy="false">)</mo></mrow></math> [ 3 3 , 11 11 , 1 1 ], where <math id="M7"><mrow><msubsup><mi>α</mi><mrow><mi>λ</mi><mn>0</mn></mrow><mo>†</mo></msubsup></mrow></math> and α λ 0 are the creation and annihilation operators of the phonons, respectively, the eigenvalues λ and eigenvectors <math id="M8"><mrow><mo>|</mo><mi>α</mi><mi>�</mi><mo>#</mo><mi>x</mi><mn>27</mn><mi>E</mi><mn>9</mn><mo>;</mo></mrow></math> of the operator <math id="M9"><mover accent="true"><mi>O</mi><mo>^</mo></mover></math> satisfy <math id="M10"><mrow><msub><mover accent="true"><mi>O</mi><mo>^</mo></mover><mi>λ</mi></msub><mo>|</mo><mi>α</mi><mi>�</mi><mo>#</mo><mi>x</mi><mn>27</mn><mi>E</mi><mn>9</mn><mo>;</mo><mo>=</mo><msub><mi>λ</mi><mi>α</mi></msub><mo>|</mo><mi>α</mi><mi>�</mi><mo>#</mo><mi>x</mi><mn>27</mn><mi>E</mi><mn>9</mn><mo>;</mo></mrow></math> , R i 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 3 ] <math display="block" id="M11"><mtable><mtr><mtd><msubsup><mi>V</mi><mrow><mi>n</mi><mi>m</mi></mrow><mtext>N</mtext></msubsup><mo>=</mo><mo>〈</mo><mi>n</mi><mo>|</mo><msub><mover accent="true"><mi>V</mi><mo stretchy="true">˜</mo></mover><mtext>N</mtext></msub><mo stretchy="false">(</mo><mi>R</mi><mn>,</mn><msub><mover accent="true"><mi>O</mi><mo>^</mo></mover><mi>λ</mi></msub><mo stretchy="false">)</mo><mo>|</mo><mi>m</mi><mo>〉</mo><mo>−</mo><msub><mi>V</mi><mtext>N</mtext></msub><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo><msub><mi>δ</mi><mrow><mi>n</mi><mn>,</mn><mi>m</mi></mrow></msub></mtd></mtr><mtr><mtd><mo>=</mo><mstyle displaystyle="true"><munder><mo>∑</mo><mi>α</mi></munder></mstyle><mo>〈</mo><mi>n</mi><mo>|</mo><mi>α</mi><mo>〉</mo><mo>〈</mo><mi>α</mi><mo>|</mo><mi>m</mi><mo>〉</mo><msub><mover accent="true"><mi>V</mi><mo stretchy="true">˜</mo></mover><mtext>N</mtext></msub><mo stretchy="false">(</mo><mi>R</mi><mn>,</mn><msub><mi>λ</mi><mi>α</mi></msub><mo stretchy="false">)</mo><mo>−</mo><msub><mi>V</mi><mtext>N</mtext></msub><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo><msub><mi>δ</mi><mrow><mi>n</mi><mn>,</mn><mi>m</mi></mrow></msub><mn>.</mn></mtd></mtr></mtable></math> The nuclear coupling potential <math id="M12"><mrow><msub><mrow><mover accent="true"><mi>V</mi><mo stretchy="true">˜</mo></mover></mrow><mtext>N</mtext></msub><mo stretchy="false">(</mo><mi>R</mi><mn>,</mn><msub><mi>λ</mi><mi>α</mi></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mi>V</mi><mtext>N</mtext></msub><mo stretchy="false">(</mo><mi>R</mi><mo>−</mo><msub><mi>λ</mi><mi>α</mi></msub><mo stretchy="false">)</mo></mrow></math> is taken up to the second-order of λ α [ 3 3 , 11 11 , 1 1 ] <math display="block" id="M13"><mrow><msub><mrow><mover accent="true"><mi>V</mi><mo stretchy="true">˜</mo></mover></mrow><mtext>N</mtext></msub><mo stretchy="false">(</mo><mi>R</mi><mn>,</mn><msub><mi>λ</mi><mi>α</mi></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mi>V</mi><mtext>N</mtext></msub><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo><mo>−</mo><mfrac><mrow><mtext>d</mtext><msub><mi>V</mi><mtext>N</mtext></msub><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo></mrow><mrow><mtext>d</mtext><mi>R</mi></mrow></mfrac><msub><mi>λ</mi><mi>α</mi></msub><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mfrac><mrow><msup><mtext>d</mtext><mn>2</mn></msup><msub><mi>V</mi><mtext>N</mtext></msub><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo></mrow><mrow><mtext>d</mtext><msup><mi>R</mi><mn>2</mn></msup></mrow></mfrac><msubsup><mi>λ</mi><mi>α</mi><mn>2</mn></msubsup><mn>,</mn></mrow></math> 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 P J 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 3 ]: <math display="block" id="M14"><mrow><msub><mi>σ</mi><mrow><mtext>fus</mtext></mrow></msub><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mi>π</mi><mrow><msup><mi>k</mi><mn>2</mn></msup></mrow></mfrac><mstyle displaystyle="true"><munder><mo>∑</mo><mi>J</mi></munder></mstyle><mo stretchy="false">(</mo><mn>2</mn><mi>J</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><msub><mi>P</mi><mi>J</mi></msub><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo><mn>,</mn></mrow></math> where <math id="M15"><mrow><mi>k</mi><mo>=</mo><msqrt><mrow><mn>2</mn><mi>μ</mi><mi>E</mi><mo>/</mo><msup><mi>ℏ</mi><mn>2</mn></msup></mrow></msqrt></mrow></math> is the wave number associated with energy E . The double folding potential 2 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 46 , 47 47 ]. However, it is difficult to explain the fusion process in a strong-density overlap region [ 48 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 21 ]: <math display="block" id="M16"><mtable columnalign="left"><mtr><mtd><msub><mi>V</mi><mtext>N</mtext></msub><mo stretchy="false">(</mo><mstyle mathvariant="bold" mathsize="normal"><mi>R</mi></mstyle><mo stretchy="false">)</mo><mo>=</mo><msub><mi>V</mi><mtext>D</mtext></msub><mo stretchy="false">(</mo><mstyle mathvariant="bold" mathsize="normal"><mi>R</mi></mstyle><mo stretchy="false">)</mo><mo>+</mo><msub><mi>V</mi><mtext>P</mtext></msub><mo stretchy="false">(</mo><mstyle mathvariant="bold" mathsize="normal"><mi>R</mi></mstyle><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mtext>          </mtext><mo>=</mo><mstyle displaystyle="true"><mrow><mo>∫</mo></mrow></mstyle><mtext>d</mtext><msub><mstyle mathvariant="bold" mathsize="normal"><mi>r</mi></mstyle><mstyle mathvariant="bold" mathsize="normal"><mn>1</mn></mstyle></msub><mtext>d</mtext><msub><mstyle mathvariant="bold" mathsize="normal"><mi>r</mi></mstyle><mstyle mathvariant="bold" mathsize="normal"><mn>2</mn></mstyle></msub><msub><mi>ρ</mi><mtext>p</mtext></msub><mo stretchy="false">(</mo><msub><mstyle mathvariant="bold" mathsize="normal"><mi>r</mi></mstyle><mstyle mathvariant="bold" mathsize="normal"><mn>1</mn></mstyle></msub><mo stretchy="false">)</mo><msub><mi>ρ</mi><mtext>t</mtext></msub><mo stretchy="false">(</mo><msub><mstyle mathvariant="bold" mathsize="normal"><mi>r</mi></mstyle><mstyle mathvariant="bold" mathsize="normal"><mn>2</mn></mstyle></msub><mo stretchy="false">)</mo><mi>g</mi><mo stretchy="false">(</mo><mo>|</mo><msub><mstyle mathvariant="bold" mathsize="normal"><mi>s</mi></mstyle><mstyle mathvariant="bold" mathsize="normal"><mn>1</mn></mstyle></msub><mo>|</mo><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mtext>           </mtext><mo>+</mo><mstyle displaystyle="true"><mrow><mo>∫</mo></mrow></mstyle><mtext>d</mtext><msub><mstyle mathvariant="bold" mathsize="normal"><mi>r</mi></mstyle><mstyle mathvariant="bold" mathsize="normal"><mn>1</mn></mstyle></msub><msub><mi>ρ</mi><mtext>p</mtext></msub><mo stretchy="false">(</mo><msub><mstyle mathvariant="bold" mathsize="normal"><mi>r</mi></mstyle><mstyle mathvariant="bold" mathsize="normal"><mn>1</mn></mstyle></msub><mo stretchy="false">)</mo><mi>v</mi><mo stretchy="false">(</mo><mo>|</mo><msub><mstyle mathvariant="bold" mathsize="normal"><mi>s</mi></mstyle><mstyle mathvariant="bold" mathsize="normal"><mn>2</mn></mstyle></msub><mo>|</mo><mo stretchy="false">)</mo><mn>,</mn></mtd></mtr></mtable></math> <math display="block" id="M17"><mrow><mi>g</mi><mo stretchy="false">(</mo><mo>|</mo><msub><mstyle mathvariant="bold" mathsize="normal"><mi>s</mi></mstyle><mstyle mathvariant="bold" mathsize="normal"><mn>1</mn></mstyle></msub><mo>|</mo><mo stretchy="false">)</mo><mo>=</mo><mn>9846</mn><mfrac><mrow><mi>exp</mi><mo stretchy="false">(</mo><mo>−</mo><mn>4</mn><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">)</mo></mrow><mrow><mn>4</mn><msub><mi>s</mi><mn>1</mn></msub></mrow></mfrac><mo>−</mo><mn>3139</mn><mfrac><mrow><mi>exp</mi><mo stretchy="false">(</mo><mo>−</mo><mn>2.5</mn><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">)</mo></mrow><mrow><mn>2.5</mn><msub><mi>s</mi><mn>1</mn></msub></mrow></mfrac><mn>,</mn></mrow></math> <math display="block" id="M18"><mrow><mi>v</mi><mo stretchy="false">(</mo><mo>|</mo><msub><mstyle mathvariant="bold" mathsize="normal"><mi>s</mi></mstyle><mstyle mathvariant="bold" mathsize="normal"><mn>2</mn></mstyle></msub><mo>|</mo><mo stretchy="false">)</mo><mo>=</mo><mn>437.05</mn><msub><mi>ρ</mi><mtext>t</mtext></msub><mo stretchy="false">(</mo><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mo>+</mo><mn>983.89</mn><msubsup><mi>ρ</mi><mtext>t</mtext><mn>2</mn></msubsup><mo stretchy="false">(</mo><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mn>.</mn></mrow></math> where <math id="M19"><mrow><mi>g</mi><mo stretchy="false">(</mo><mo>|</mo><msub><mstyle mathvariant="bold" mathsize="normal"><mi>s</mi></mstyle><mstyle mathvariant="bold" mathsize="normal"><mn>1</mn></mstyle></msub><mstyle mathvariant="bold" mathsize="normal"><mo>=</mo><mi>R</mi></mstyle><mo>−</mo><msub><mstyle mathvariant="bold" mathsize="normal"><mi>r</mi></mstyle><mstyle mathvariant="bold" mathsize="normal"><mn>1</mn></mstyle></msub><mo>+</mo><msub><mstyle mathvariant="bold" mathsize="normal"><mi>r</mi></mstyle><mstyle mathvariant="bold" mathsize="normal"><mn>2</mn></mstyle></msub><mo>|</mo><mo stretchy="false">)</mo></mrow></math> denotes the direct interaction between a nucleon in the target nucleus and a nucleon in the projectile nuclei, and <math id="M20"><mrow><mi>v</mi><mo stretchy="false">(</mo><mo>|</mo><msub><mstyle mathvariant="bold" mathsize="normal"><mi>s</mi></mstyle><mstyle mathvariant="bold" mathsize="normal"><mn>2</mn></mstyle></msub><mstyle mathvariant="bold" mathsize="normal"><mo>=</mo><mi>R</mi></mstyle><mo>−</mo><msub><mstyle mathvariant="bold" mathsize="normal"><mi>r</mi></mstyle><mstyle mathvariant="bold" mathsize="normal"><mn>1</mn></mstyle></msub><mo>|</mo><mo stretchy="false">)</mo></mrow></math> 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 21 ]. The Coulomb potential employed in the calculations was the double-folding integral of the proton-proton Coulomb interaction: <math display="block" id="M21"><mrow><msub><mi>V</mi><mtext>C</mtext></msub><mo stretchy="false">(</mo><mstyle mathvariant="bold" mathsize="normal"><mi>R</mi></mstyle><mo stretchy="false">)</mo><mo>=</mo><mstyle displaystyle="true"><mrow><mo>∫</mo></mrow></mstyle><mtext>d</mtext><msub><mstyle mathvariant="bold" mathsize="normal"><mi>r</mi></mstyle><mstyle mathvariant="bold" mathsize="normal"><mn>1</mn></mstyle></msub><mtext>d</mtext><msub><mstyle mathvariant="bold" mathsize="normal"><mi>r</mi></mstyle><mstyle mathvariant="bold" mathsize="normal"><mn>2</mn></mstyle></msub><mfrac><mrow><msup><mi>e</mi><mn>2</mn></msup></mrow><mrow><mo>|</mo><msub><mstyle mathvariant="bold" mathsize="normal"><mi>s</mi></mstyle><mstyle mathvariant="bold" mathsize="normal"><mn>1</mn></mstyle></msub><mo>|</mo></mrow></mfrac><msub><mi>ρ</mi><mrow><mtext>pp</mtext></mrow></msub><mo stretchy="false">(</mo><msub><mstyle mathvariant="bold" mathsize="normal"><mi>r</mi></mstyle><mstyle mathvariant="bold" mathsize="normal"><mn>1</mn></mstyle></msub><mo stretchy="false">)</mo><msub><mi>ρ</mi><mrow><mtext>tp</mtext></mrow></msub><mo stretchy="false">(</mo><msub><mstyle mathvariant="bold" mathsize="normal"><mi>r</mi></mstyle><mstyle mathvariant="bold" mathsize="normal"><mn>2</mn></mstyle></msub><mo stretchy="false">)</mo><mn>,</mn></mrow></math> 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 49 ]: <math display="block" id="M22"><mrow><msub><mi>ρ</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><msub><mi>ρ</mi><mrow><mn>0</mn><mi>i</mi></mrow></msub></mrow><mrow><mn>1</mn><mo>+</mo><mi>exp</mi><mo stretchy="false">(</mo><mfrac><mrow><mi>r</mi><mo>−</mo><msub><mi>C</mi><mi>i</mi></msub></mrow><mrow><msub><mi>a</mi><mi>i</mi></msub></mrow></mfrac><mo stretchy="false">)</mo></mrow></mfrac><mn>,</mn></mrow></math> where ρ i , C i , and a i indicate the density, half-density radius, and diffuseness of the neutrons ( i =n) and protons ( i =p), respectively. ρ 0 i 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 i 0 , C i , and a i in Eq. (9) are fixed during the fusion process. The non-frozen density approximation 2 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 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 38 , 43 43 , 50 50 , 51 51 ]. In [ 43 43 ], Reichstein and Malik introduced a special nonfrozen density approximation (NFDA), the 16 O + 16 O fusion reaction. A similar approach was recently successfully employed to describe the elastic scattering and fusion data for the 12 C + 12 C reaction [ 44 44 , 45 45 ]. This approach introduces the distance dependence R to the density parameters C and a as follows: <math display="block" id="M23"><mrow><mi>C</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mo>{</mo><mrow><mtable><mtr><mtd><mrow><mi>C</mi><mn>,</mn><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mi>R</mi><mo>></mo><msub><mi>R</mi><mtext>c</mtext></msub></mrow></mtd></mtr><mtr><mtd><mrow><msub><mi>C</mi><mrow><mtext>cn</mtext></mrow></msub><mi>exp</mi><mo stretchy="false">[</mo><mi>ln</mi><mo stretchy="false">(</mo><mfrac><mi>C</mi><mrow><msub><mi>C</mi><mrow><mtext>cn</mtext></mrow></msub></mrow></mfrac><mo stretchy="false">)</mo><mo>⋅</mo><mfrac><mrow><msup><mi>R</mi><mn>2</mn></msup></mrow><mrow><msubsup><mi>R</mi><mtext>c</mtext><mn>2</mn></msubsup></mrow></mfrac><mo stretchy="false">]</mo><mn>,</mn><mtext> </mtext><mtext> </mtext><mi>R</mi><mo>≤</mo><msub><mi>R</mi><mtext>c</mtext></msub></mrow></mtd></mtr></mtable></mrow></mrow></mrow></math> 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 37 , 38 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 38 ] <math display="block" id="M24"><mrow><mi>Φ</mi><mo stretchy="false">(</mo><mi>R</mi><mn>,</mn><msub><mi>λ</mi><mi>α</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mrow><mo>{</mo><mrow><mtable><mtr><mtd><mrow><mi>exp</mi><mo stretchy="false">[</mo><mo>−</mo><msup><mrow><mo stretchy="false">(</mo><mi>R</mi><mo>−</mo><msub><mi>R</mi><mtext>d</mtext></msub><mo>−</mo><msub><mi>λ</mi><mi>α</mi></msub><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo>/</mo><mn>2</mn><msubsup><mi>a</mi><mtext>d</mtext><mn>2</mn></msubsup><mo stretchy="false">]</mo><mn>,</mn></mrow></mtd><mtd><mrow><mi>R</mi><mo>≤</mo><msub><mi>R</mi><mtext>d</mtext></msub><mo>+</mo><msub><mi>λ</mi><mi>α</mi></msub><mn>,</mn></mrow></mtd></mtr><mtr><mtd><mrow><mn>1,</mn><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext></mrow></mtd><mtd><mrow><mi>R</mi><mo>></mo><msub><mi>R</mi><mtext>d</mtext></msub><mo>+</mo><msub><mi>λ</mi><mi>α</mi></msub><mn>,</mn></mrow></mtd></mtr></mtable></mrow></mrow></mrow></math> 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 37 , 38 38 ]: <math display="block" id="M25"><mtable><mtr><mtd><msub><mover accent="true"><mi>V</mi><mo stretchy="true">˜</mo></mover><mtext>N</mtext></msub><mo stretchy="false">(</mo><mi>R</mi><mn>,</mn><msub><mi>λ</mi><mi>α</mi></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mi>V</mi><mtext>N</mtext></msub><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo><mo>−</mo><mo stretchy="false">[</mo><mfrac><mrow><mtext>d</mtext><msub><mi>V</mi><mtext>N</mtext></msub><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo></mrow><mrow><mtext>d</mtext><mi>R</mi></mrow></mfrac><msub><mi>λ</mi><mi>α</mi></msub></mtd></mtr><mtr><mtd><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mfrac><mrow><msup><mtext>d</mtext><mn>2</mn></msup><msub><mi>V</mi><mtext>N</mtext></msub><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo></mrow><mrow><mtext>d</mtext><msup><mi>R</mi><mn>2</mn></msup></mrow></mfrac><msubsup><mi>λ</mi><mi>α</mi><mn>2</mn></msubsup><mo stretchy="false">]</mo><mi>Φ</mi><mo stretchy="false">(</mo><mi>R</mi><mn>,</mn><msub><mi>λ</mi><mi>α</mi></msub><mo stretchy="false">)</mo><mn>.</mn></mtd></mtr></mtable></math> 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. results and discussions 1 Considering the medium-heavy mass system of 64 Ni + 64 Ni and medium-light mass system of 24 Mg + 30 Si as examples, the potentials and fusion cross-sections obtained by the aforementioned non-frozen process are discussed in the following sections. Before presenting the detailed theoretical results, we clarify the parameters used in the calculations. The densities used to calculate the double-folding potential are listed in Table 1 Table 1 . The input parameters for the coupling strengths in the CC calculations for 64 Ni + 64 Ni and 24 Mg + 30 Si systems were obtained from Ref. [ 38 38 ] and are tabulated in Table 2 Table 2 . The damping radius R d and diffuseness a d parameters used in Eq. (11) are 1.298 fm and 1.25 fm for the 64 Ni + 64 Ni fusion system, and 1.430 fm and 1.25 fm for the 24 Mg + 30 Si fusion system [ 38 38 ]. C i (fm) and a i (fm) are the half-density radius and diffuseness of the neutron ( i = n) and proton ( i = p) used in Eq.(9). The last column presents the corresponding references where the parameters were obtained from. 1 1 Nucleus 1 1 C p 1 1 a p 1 1 C n 1 1 a n 1 1 Refences 1 1 24 Mg 1 1 3.032 1 1 0.52 1 1 3.032 1 1 0.52 1 1 [ 52 52 ] 1 1 30 Si 1 1 3.130 1 1 0.48 1 1 3.130 1 1 0.48 1 1 [ 52 52 ] 1 1 54 Fe 1 1 4.134 1 1 0.51 1 1 4.058 1 1 0.52 1 1 [ 53 53 ] 1 1 64 Ni 1 1 4.260 1 1 0.58 1 1 4.340 1 1 0.53 1 1 [ 31 31 ] 1 1 128 Ba 1 1 5.580 1 1 0.50 1 1 5.570 1 1 0.53 1 1 [ 53 53 ] λ π denotes the multipolarity and parity of a state. E ex denotes the excitation energy of a state. β c and β n denote the deformation parameters for the Coulomb and nuclear coupling strengths, respectively. N ph in the last column indicates the number of phonons included in the calculations. 1 1 Nucleus 1 1 λ π 1 1 E ex (MeV) 1 1 β c 1 1 β n 1 1 N ph 1 1 24 Mg 1 1 2 + 1 1 1.369 1 1 0.608 1 1 0.460 1 1 1 1 1 30 Si 1 1 2 + 1 1 2.235 1 1 0.330 1 1 0.330 1 1 1 1 1   1 1 3 - 1 1 5.497 1 1 0.275 1 1 0.275 1 1 1 1 1 64 Ni 1 1 2 + 1 1 1.346 1 1 0.165 1 1 0.185 1 1 2 1 1   1 1 3 - 1 1 3.560 1 1 0.193 1 1 0.200 1 1 1 the medium-heavy mass system 64 Ni + 64 Ni 2 First, the density distribution of the FDA and NFDA after two colliding nuclei overlap is presented in Fig. 1 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 64 Ni + 64 Ni 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 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 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. Here, we provide a detailed comparison of the potentials demonstrated in Fig. 2 Fig. 2 for the 64 Ni + 64 Ni 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) 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) 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) 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. The total potentials and fusion cross-sections are shown in Fig. 3 Fig. 3 . The commonly used potentials in heavy-ion fusion reactions, such as the Akyüz-Winther (AW) potential [ 39 39 ], are also presented in Fig. 3(a) 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. In Fig. 3(b) Fig. 3(b) , the calculated fusion cross sections based on the CC model are compared with the experimental data [ 54 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) 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 24 Mg + 30 Si 2 Figure 4 Figure 4 presents the obtained total potential and fusion cross sections for the medium-light mass system 24 Mg + 30 Si. Similar to the 64 Ni + 64 Ni 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) 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) 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. 4(b) . In comparison with the medium-light mass system 24 Mg + 30 Si, the density rearrangement of the 64 Ni + 64 Ni 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 64 Ni + 64 Ni fusion system. However, in the relatively light mass system 24 Mg + 30 Si, 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 12 C + 12 C at low energies, which strongly affects the estimation of astrophysical reaction rates. Summary 1 The fusion hindrance phenomenon observed at deep sub-barrier energies was investigated using a double folding model based on the non-frozen density approximation. In this non-frozen method, the density distribution transformation from the projectile and target nuclei to the compound nucleus is described by introducing the distance dependence to the half-density radius and diffuseness parameters. When the density of the overlapping region reaches the central density of the compound nuclei, the projectile and target density distributions begin to change. The enlarged half-density radius reduces the central density of the colliding nucleus according to the normalization condition. At the initial stage of the density rearrangement, the increasing surface density of the colliding nuclei plays a dominant role in the overlapping region. The slightly higher overlap density of the non-frozen process causes the potential to have a shallower pocket compared to that of the frozen potential. As the distance decreases, the non-frozen fusion process exhibits a relatively weak density overlap, and the corresponding potential is significantly more attractive than the frozen potential. The CC effects have a significant influence on heavy ion fusion reactions. In the non-frozen picture, the CC effects decrease owing to the density rearrangement, and a damping factor is employed to simulate this process. Based on the non-frozen approximation with the damping factor, the calculated fusion cross sections and astrophysical S factors for 64 Ni + 64 Ni and 24 Mg + 30 Si are both in good agreement with the experimental data. For a relatively heavy mass fusion system, such as the 64 Ni + 64 Ni reaction, the non-frozen process with the damping of the CC strength causes a drastic reduction in the fusion cross-section, and a clear maximum is exhibited in the astrophysical S factor. In the medium-light mass system 24 Mg + 30 Si, the damping effect is relatively weak, and the predicted S factor demonstrates that there is no maximum at low collision energies. It would be noteworthy to extend this method to astrophysical fusion systems, such as the 12 C + 12 C reaction. For the 12 C + 12 C fusion reaction at low energies, the fusion process can be considered a superposition of the non-resonant background and an additional contribution of resonance [ 56 56 - 59 59 56-59 ]. This method can be helpful in understanding the fusion process of non-resonant effects at astrophysical energies for 12 C + 12 C reactions. References H. Esbensen and S. Landowne , Higher-order coupling effects in low energy heavy-ion fusion reactions . Phys. Rev. C 35 , 2090 ( 1987 ). doi: 10.1103/PhysRevC.35.2090 http://doi.org/10.1103/PhysRevC.35.2090 Higher-order coupling effects in low energy heavy-ion fusion reactions A. B. Balantekin and N. Takigawa , Quantum tunneling in nuclear fusion . Rev. Mod. Phys. 70 , 77 ( 1998 ). doi: 10.1103/RevModPhys.70.77 http://doi.org/10.1103/RevModPhys.70.77 Quantum tunneling in nuclear fusion K. Hagino and N. Takigawa , Subbarrier fusion reactions and many-particle quantum tunneling . Prog. Theor. Phys. 128 , 1061 ( 2012 ). doi: 10.1143/PTP.128.1061 http://doi.org/10.1143/PTP.128.1061 Subbarrier fusion reactions and many-particle quantum tunneling J. O. Newton , C. R. Morton , M. Dasgupta et al. , Experimental barrier distributions for the fusion of 12 C, 16 O, 28 Si, and 35 Cl with 92 Zr and coupled-channels analyses . Phys. Rev. C 64 , 064608 ( 2001 ). doi: 10.1103/PhysRevC.64.064608 http://doi.org/10.1103/PhysRevC.64.064608 Experimental barrier distributions for the fusion of 12C, 16O, 28Si, and 35Cl with 92Zr and coupled-channels analyses K. Hagino , N. Takigawa , M. Dasgupta , et al. , Validity of the linear coupling approximation in heavy-ion fusion reactions at sub-barrier energies . Phys. Rev. C 55 276 ( 1997 ). doi: 10.1103/PhysRevC.55.276 http://doi.org/10.1103/PhysRevC.55.276 Validity of the linear coupling approximation in heavy-ion fusion reactions at sub-barrier energies C.J. Lin , H.M. Jia , H.Q. Zhang et al. , Systematic study of the surface properties of the nuclear potential with high precision large-angle quasi-elastic scatterings . Phys. Rev. C 79 , 064603 ( 2009 ). doi: 10.1103/PhysRevC.79.064603 http://doi.org/10.1103/PhysRevC.79.064603 Systematic study of the surface properties of the nuclear potential with high precision large-angle quasi-elastic scatterings F. Niu , P.-H. Chen , H.-G. Cheng et al. , Multinucleon transfer dynamics in nearly symmetric nuclear reactions . Nucl. Sci. Tech. 31 , 59 ( 2020 ). doi: 10.1007/s41365-020-00770-1 http://doi.org/10.1007/s41365-020-00770-1 Multinucleon transfer dynamics in nearly symmetric nuclear reactions P.-H. Chen , F. Niu , Y.-F. Guo , Z.-Q. Feng , Nuclear dynamics in multinucleon transfer reactions near Coulomb barrier energies . Nucl. Sci. Tech. 29 , 185 ( 2018 ). doi: 10.1007/s41365-018-0521-y http://doi.org/10.1007/s41365-018-0521-y Nuclear dynamics in multinucleon transfer reactions near Coulomb barrier energies C. H. Dasso and S. Landowne , CCFUS — A simplified coupled-channel code for calculation of fusion cross sections in heavy-ion reactions . Comput. Phys. Commun. 46 , 187 ( 1987 ). doi: 10.1016/0010-4655(87)90045-2 http://doi.org/10.1016/0010-4655(87)90045-2 CCFUS — A simplified coupled-channel code for calculation of fusion cross sections in heavy-ion reactions J. Fernández-Niello , C. H. Dasso , S. Landowne , CCDEF - A simplified coupled-channel code for fusion cross sections including static nuclear deformations . Comput. Phys. Commun. 54 , 409 ( 1989 ). doi: 10.1016/0010-4655(89)90100-8 http://doi.org/10.1016/0010-4655(89)90100-8 CCDEF - A simplified coupled-channel code for fusion cross sections including static nuclear deformations K. Hagino , N. Rowley , A. T. Kruppa , A program for coupled-channel calculations with all order couplings for heavy-ion fusion reactions . Comput. Phys. Commun. 123 , 143 ( 1999 ). doi: 10.1016/S0010-4655(99)00243-X http://doi.org/10.1016/S0010-4655(99)00243-X A program for coupled-channel calculations with all order couplings for heavy-ion fusion reactions M. Hosamani , A. Vinayak , N. Badiger , Investigating the effect of entrance channel mass asymmetry on fusion reactions using the Skyrme energy density formalism . Nucl. Sci. Tech. 31 , 89 ( 2020 ). doi: 10.1007/s41365-020-00802-w http://doi.org/10.1007/s41365-020-00802-w Investigating the effect of entrance channel mass asymmetry on fusion reactions using the Skyrme energy density formalism C.L. Jiang , H. Esbensen , K.E. Rehm et al. , Unexpected behavior of heavy-Ion fusion cross sections at extreme sub-barrier energies . Phys. Rev. Lett. 89 , 052701 ( 2002 ). doi: 10.1103/PhysRevLett.89.052701 http://doi.org/10.1103/PhysRevLett.89.052701 Unexpected behavior of heavy-Ion fusion cross sections at extreme sub-barrier energies B.B. Back , H. Esbensen , C.L. Jiang et al. , Recent developments in heavy-ion fusion reactions . Rev. Mod. Phys. 86 , 317 ( 2014 ). doi: 10.1103/RevModPhys.86.317 http://doi.org/10.1103/RevModPhys.86.317 Recent developments in heavy-ion fusion reactions M. Dasgupta , D.J. Hinde , A. Diaz-Torres et al. , Beyond the coherent coupled channels description of nuclear fusion . Phys. Rev. Lett. 99 , 192701 ( 2007 ). doi: 10.1103/PhysRevLett.99.192701 http://doi.org/10.1103/PhysRevLett.99.192701 Beyond the coherent coupled channels description of nuclear fusion K. Hagino , N. Rowley , M. Dasgupta , Fusion cross sections at deep sub-barrier energies . Phys. Rev. C 67 , 054603 ( 2003 ). doi: 10.1103/PhysRevC.67.054603 http://doi.org/10.1103/PhysRevC.67.054603 Fusion cross sections at deep sub-barrier energies V.V. Sargsyan , Z. Kanokov , G.G. Adamian et al. , Capture process in nuclear reactions with a quantum master equation . Phys. Rev. C 80 , 034606 ( 2009 ). doi: 10.1103/PhysRevC.80.034606 http://doi.org/10.1103/PhysRevC.80.034606 Capture process in nuclear reactions with a quantum master equation V. Yu Denisov , Nucleus-nucleus potential with shell-correction contribution . Phys. Rev. C 91 , 024603 ( 2015 ). doi: 10.1103/PhysRevC.91.024603 http://doi.org/10.1103/PhysRevC.91.024603 Nucleus-nucleus potential with shell-correction contribution C.L. Jiang , B.B. Back , K.E. Rehm et al. , Heavy-ion fusion reactions at extreme subbarrier energies . Eur. Phys. J. A 57 , 235 ( 2021 ). doi: 10.1140/epja/s10050-021-00536-2 http://doi.org/10.1140/epja/s10050-021-00536-2 Heavy-ion fusion reactions at extreme subbarrier energies V. Yu. Denisov , Multidimensional harmonic oscillator model of subbarrier fusion . Eur. Phys. J. A 58 , 91 ( 2022 ). doi: 10.1140/epja/s10050-022-00746-2 http://doi.org/10.1140/epja/s10050-022-00746-2 Multidimensional harmonic oscillator model of subbarrier fusion K.-X. Cheng , C. Xu , C.-W. Ma et al. , Pauli blocking potential applied to heavy-ion fusion reactions . Chin. Phys. C 46 , 024105 ( 2022 ). doi: 10.1088/1674-1137/ac3749 http://doi.org/10.1088/1674-1137/ac3749 Pauli blocking potential applied to heavy-ion fusion reactions D. Boilley , B. Cauchois , H.L. Lü et al. , How accurately can we predict synthesis cross sections of superheavy elements? Nucl. Sci. Tech. 29 , 172 ( 2018 ). doi: 10.1007/s41365-018-0509-7 http://doi.org/10.1007/s41365-018-0509-7 How accurately can we predict synthesis cross sections of superheavy elements? Z.-Q. Feng , Nuclear dynamics and particle production near threshold energies in heavy-ion collisions . Nucl. Sci. Tech. 29 , 40 ( 2018 ). doi: 10.1007/s41365-018-0379-z http://doi.org/10.1007/s41365-018-0379-z Nuclear dynamics and particle production near threshold energies in heavy-ion collisions C.L. Jiang , B.B. Back , H. Esbensen et al. , Origin and consequences of 12 C + 12 C fusion resonances at deep sub-barrier energies . Phys. Rev. Lett. 110 , 072701 ( 2013 ). doi: 10.1103/PhysRevLett.110.072701 http://doi.org/10.1103/PhysRevLett.110.072701 Origin and consequences of 12C + 12C fusion resonances at deep sub-barrier energies C.L. Jiang , D. Santiago-Gonzalez , S. Almaraz-Calderon et al. , Reaction rate for carbon burning in massive stars . Phys. Rev. C 97 , 012801(R) 2018 . doi: 10.1103/PhysRevC.97.012801 http://doi.org/10.1103/PhysRevC.97.012801 Reaction rate for carbon burning in massive stars N.T. Zhang , X.Y. Wang , D. Tudor et al. , Constraining the 12 C + 12 C astrophysical S-factors with the 12 C + 13 C measurements at very low energies . Phys. Lett. B 801 , 135170 ( 2020 ). doi: 10.1016/j.physletb.2019.135170 http://doi.org/10.1016/j.physletb.2019.135170 Constraining the 12C + 12C astrophysical S-factors with the 12C + 13C measurements at very low energies T.-P. Luo , P.-W. Wen , C.-J. Lin et al. , Bayesian analysis on non-resonant behavior of 12 C + 12 C fusion reaction at sub-barrier energies . Chin. Phys. C 46 , 064105 ( 2022 ). doi: 10.1088/1674-1137/ac5587 http://doi.org/10.1088/1674-1137/ac5587 Bayesian analysis on non-resonant behavior of 12C + 12C fusion reaction at sub-barrier energies V.V. Sargyan , G.G. Adamian , N.V. antonenko et al. , Constraints on the appearance of a maximum in astrophysical S-factor . Phys. Lett. B 824 , 136792 ( 2022 ). doi: 10.1016/j.physletb.2021.136792 http://doi.org/10.1016/j.physletb.2021.136792 Constraints on the appearance of a maximum in astrophysical S-factor X.-D. Tang , S.-B. Ma , X. Fang et al. , An efficient method for mapping the 12 C+ 12 C molecular resonances at low energies . Nucl. Sci. Tech. 30 , 126 ( 2019 ). doi: 10.1007/s41365-019-0652-9 http://doi.org/10.1007/s41365-019-0652-9 An efficient method for mapping the 12C+12C molecular resonances at low energies Ş. Mişicu , H. Esbensen , Hindrance of heavy-ion fusion due to nuclear incompressibility . Phys. Rev. Lett. 96 , 112701 ( 2006 ). doi: 10.1103/PhysRevLett.96.112701 http://doi.org/10.1103/PhysRevLett.96.112701 Hindrance of heavy-ion fusion due to nuclear incompressibility Ş. Mişicu , H. Esbensen , Signature of shallow potentials in deep sub-barrier fusion reactions . Phys. Rev. C 75 , 034606 ( 2007 ). doi: 10.1103/PhysRevC.75.034606 http://doi.org/10.1103/PhysRevC.75.034606 Signature of shallow potentials in deep sub-barrier fusion reactions H. Esbensen , Ş. Mişicu , Hindrance of 16 O + 208 Pb fusion at extreme subbarrier energies . Phys. Rev. C 76 , 054609 ( 2007 ). doi: 10.1103/PhysRevC.76.054609 http://doi.org/10.1103/PhysRevC.76.054609 Hindrance of 16O + 208Pb fusion at extreme subbarrier energies C. Simenel , A.S. Umar , K. Godbey et al. , How the Pauli exclusion principle affects fusion of atomic nuclei . Phys. Rev. C 95 , 031601(R) ( 2017 ). doi: 10.1103/PhysRevC.95.031601 http://doi.org/10.1103/PhysRevC.95.031601 How the Pauli exclusion principle affects fusion of atomic nuclei A.S. Umar , C. Simennel , K. Godbey , Pauli energy contribution to the nucleus-nucleus interaction . Phys. Rev. C 104 , 034619 ( 2021 ). doi: 10.1103/PhysRevC.104.034619 http://doi.org/10.1103/PhysRevC.104.034619 Pauli energy contribution to the nucleus-nucleus interaction K.X. Cheng , C. Xu , Pauli blocking effects in α -induced fusion reactions . Phys. Rev. C 99 , 014607 ( 2019 ). doi: 10.1103/PhysRevC.99.014607 http://doi.org/10.1103/PhysRevC.99.014607 Pauli blocking effects in α-induced fusion reactions K.X. Cheng , C. Xu , Pauli blocking effects in nα -nucleus-induced fusion reactions . Phys. Rev. C 102 , 014619 ( 2020 ). doi: 10.1103/PhysRevC.102.014619 http://doi.org/10.1103/PhysRevC.102.014619 Pauli blocking effects in nα-nucleus-induced fusion reactions T. Ichikawa , K. Hagino , A. Iwamoto , Signature of smooth transition from sudden to adiabatic states in heavy-ion fusion reactions at deep sub-barrier energies . Phys. Rev. Lett. 103 , 202701 ( 2009 ). doi: 10.1103/PhysRevLett.103.202701 http://doi.org/10.1103/PhysRevLett.103.202701 Signature of smooth transition from sudden to adiabatic states in heavy-ion fusion reactions at deep sub-barrier energies T. Ichikawa , Systematic investigations of deep sub-barrier fusion reactions using an adiabatic approach . Phys. Rev. C 92 , 064604 ( 2015 ). doi: 10.1103/PhysRevC.92.064604 http://doi.org/10.1103/PhysRevC.92.064604 Systematic investigations of deep sub-barrier fusion reactions using an adiabatic approach A. Winther , Dissipation, polarization and fluctuation in grazing heavy-ion collisions and the boundary to the chaotic regime . Nucl. Phys. A 594 , 203 ( 1995 ). doi: 10.1016/0375-9474(95)00374-A http://doi.org/10.1016/0375-9474(95)00374-A Dissipation, polarization and fluctuation in grazing heavy-ion collisions and the boundary to the chaotic regime G. Röpke , P. Schuck , Y. Funaki et al. , Nuclear clusters bound to doubly magic nuclei: The case of 212 Po . Phys. Rev. C 90 , 034304 ( 2014 ). doi: 10.1103/PhysRevC.90.034304 http://doi.org/10.1103/PhysRevC.90.034304 Nuclear clusters bound to doubly magic nuclei: The case of212Po C. Xu , G. Röpke , P. Schuck et al. , α -cluster formation and decay in the quartetting wave function approach . Phys. Rev. C 95 , 061306(R) ( 2017 ). doi: 10.1103/PhysRevC.95.061306 http://doi.org/10.1103/PhysRevC.95.061306 α-cluster formation and decay in the quartetting wave function approach S. Yang , C. Xu , G. Röpke et al. , α decay to a doubly magic core in the quartetting wave function approach . Phys. Rev. C 101 , 024316 ( 2020 ). doi: 10.1103/PhysRevC.101.024316 http://doi.org/10.1103/PhysRevC.101.024316 α decay to a doubly magic core in the quartetting wave function approach I. Reichstein , F.B. Malik , Dependence of 16 O + 16 O potential on the density ansatz . Phys. Lett. B 37 , 344 ( 1971 ). doi: 10.1016/0370-2693(71)90197-3 http://doi.org/10.1016/0370-2693(71)90197-3 Dependence of 16O + 16O potential on the density ansatz L.H. Chien , D. T. Khoa , D.C. Cuong et al. , Consistent mean-field description of the 12 C + 12 C optical potential at low energies and the astrophysical S factor . Phys. Rev. C 98 , 064604 ( 2018 ). doi: 10.1103/PhysRevC.98.064604 http://doi.org/10.1103/PhysRevC.98.064604 Consistent mean-field description of the 12C + 12C optical potential at low energies and the astrophysical S factor D.T. Khoa , L.H. Chien , D.C. Cuong et al. , Mean-field description of heavy-ion scattering at low energies and fusion . Nucl. Sci. Tech. 29 , 183 ( 2018 ). doi: 10.1007/s41365-018-0517-7 http://doi.org/10.1007/s41365-018-0517-7 Mean-field description of heavy-ion scattering at low energies and fusion G.R. Satchler , W.G. Love , Folding model potentials from realistic interactions for heavy-ion scattering . Phys. Rep. 55 , 183 ( 1979 ). doi: 10.1016/0370-1573(79)90081-4 http://doi.org/10.1016/0370-1573(79)90081-4 Folding model potentials from realistic interactions for heavy-ion scattering D.T. Khoa , α -nucleus optical potential in the double-folding model . Phys. Rev. C 63 , 034007 ( 2001 ). doi: 10.1103/PhysRevC.63.034007 http://doi.org/10.1103/PhysRevC.63.034007 α-nucleus optical potential in the double-folding model I.I. Gontcher , D.J. Hinde , M. Dasgupta et al. , Double folding nucleus-nucleus potential applied to heavy-ion fusion reactions . Phys. Rev. C 69 , 024610 ( 2004 ). doi: 10.1103/PhysRevC.69.024610 http://doi.org/10.1103/PhysRevC.69.024610 Double folding nucleus-nucleus potential applied to heavy-ion fusion reactions A. Bohr , B.R. Mottelson , Nuclear Structure . World Scientific , Singapore , 1988 , Vol.1 . Nuclear Structure V. Yu. Denisov , Multidimensional Model of Cluster Radioactivity . Phys. Rev. C 88 , 044608 ( 2013 ). doi: 10.1103/PhysRevC.88.044608 http://doi.org/10.1103/PhysRevC.88.044608 Multidimensional Model of Cluster Radioactivity V. Yu. Denisov , Nucleus-nucleus potential with shell correction contribution and deep sub-barrier fusion of heavy nuclei . Phys. Rev. C 89 , 044604 ( 2014 ). doi: 10.1103/PhysRevC.89.044604 http://doi.org/10.1103/PhysRevC.89.044604 Nucleus-nucleus potential with shell correction contribution and deep sub-barrier fusion of heavy nuclei C.L. Jiang , A.M. Stefanini , H. Esbensen et al. , Fusion hindrance for a positive-Q-value system 24 Mg + 30 Si . Phys. Rev. Lett. 113 , 022701 ( 2014 ). doi: 10.1103/PhysRevLett.113.022701 http://doi.org/10.1103/PhysRevLett.113.022701 Fusion hindrance for a positive-Q-value system 24Mg + 30Si W.M. Seif , H. Mansour , Systematics of nucleon density distributions and neutron skin of nuclei . Int. J. Mod. Phys. E 24 , 1550083 ( 2015 ). doi: 10.1142/S0218301315500834 http://doi.org/10.1142/S0218301315500834 Systematics of nucleon density distributions and neutron skin of nuclei C.L. Jiang , K.E. Rehm , R.V.F. Janssens et al. , Influence of nuclear structure on sub-barrier hindrance in Ni + Ni fusion . Phys. Rev. Lett. 93 , 012701 ( 2004 ). doi: 10.1103/PhysRevLett.93.012701 http://doi.org/10.1103/PhysRevLett.93.012701 Influence of nuclear structure on sub-barrier hindrance in Ni + Ni fusion A. Morsad , J.J. Kolata , R.J. Tighe et al. , Sub-barrier fusion of 28,30 Si with 24,26 Mg . Phys. Rev. C 41 , 988 ( 1990 ). doi: 10.1103/PhysRevC.41.988 http://doi.org/10.1103/PhysRevC.41.988 Sub-barrier fusion of28,30Si with 24,26Mg E.F. Aguilera , P. Rosales , E. Martinez-Quiroz et al. , New γ -ray measurements for 12 C + 12 C sub-Coulomb fusion: Toward data unification . Phys. Rev. C 73 , 064601 ( 2006 ). doi: 10.1103/PhysRevC.73.064601 http://doi.org/10.1103/PhysRevC.73.064601 New γ-ray measurements for 12C + 12C sub-Coulomb fusion: Toward data unification R.L. Cooper , A.W. Steiner , E.F. Brown , Possible resonances in the 12 C + 12 C fusion rate and superburst ignition . Astrophysical Journal 702 , 660 ( 2009 ). doi: 10.1088/0004-637X/702/1/660 http://doi.org/10.1088/0004-637X/702/1/660 Possible resonances in the 12C + 12C fusion rate and superburst ignition A. Tumino , C. Spitaleri , M. La Cognata et al. , An increase in the 12 C + 12 C fusion rate from resonances at astrophysical energies . Nature 557 , 687 ( 2018 ), doi: 10.1038/s41586-018-0149-4 http://doi.org/10.1038/s41586-018-0149-4 An increase in the 12C + 12C fusion rate from resonances at astrophysical energies Y. Taniguchi , M. Kimura , 12 C + 12 C fusion S-factor from a full-microscopic nuclear model . Phys. Lett. B 823 , 136790 ( 2021 ). doi: 10.1016/j.physletb.2021.136790 http://doi.org/10.1016/j.physletb.2021.136790 12C + 12C fusion S-factor from a full-microscopic nuclear model All authors contributed to the study conception and design. Material preparation, data collection and analysis were performed by Kai-Xuan Cheng, Jie Pu and Chun-Wang Ma. The first draft of the manuscript was written by Kai-Xuan Cheng and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript. 10.1007/s41365-022-01114-x This work was supported by the National Natural Science Foundation of China (Nos. 12105080, 12105079, and 11975091) and the Program for Innovative Research Team (in Science and Technology) in University of Henan Province, China (No. 21IRTSTHN011). 2022-06-13 2022-08-24 2022-08-28 2022-10-14 2022-11-22 © China Science Publishing & Media Ltd. (Science Press), Shanghai Institute of Applied Physics, the Chinese Academy of Sciences, ChineseNuclear Society and Springer Nature Singapore Pte Ltd. 2022 This is a post-peer-review, pre-copyedit version of an article published in Nuclear Science and Techniques. The final authenticated version is available online at: http://dx.doi.org/10.1007/s41365-022-01114-x http://dx.doi.org/10.1007/s41365-022-01114-x . Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. 2022 Nuclear Science and Techniques 10 33