Mean-field description of heavy-ion scattering at low energies and fusion

Mean-field description of heavy-ion scattering at low energies and fusion 1 first-author 1 2 corresp Corresponding author: lhchien@hcmus.edu.vn; Corresponding author: lhchien@hcmus.edu.vn; lhchien@hcmus.edu.vn 1 1 Institute for Nuclear Science and Technology, VINATOM, 179 Hoang Quoc Viet, Hanoi, Vietnam Institute for Nuclear Science and Technology, VINATOM, 179 Hoang Quoc Viet, Hanoi, Vietnam Department of Nuclear Physics and Nuclear Engineering, Faculty of Physics and Engineering Physics, University of Science, VNU-HCM, 227 Nguyen Van Cu Street, District 5, Ho Chi Minh City, Vietnam Department of Nuclear Physics and Nuclear Engineering, Faculty of Physics and Engineering Physics, University of Science, VNU-HCM, 227 Nguyen Van Cu Street, District 5, Ho Chi Minh City, Vietnam The nuclear mean-field potential built up during the 12 C+ 12 C and 16 O+ 16 O collisions at low energies relevant for the carbon- and oxygen-burning processes is constructed within the double-folding model (DFM) using the realistic ground-state densities of 12 C and 16 O, and CDM3Yn density-dependent nucleon-nucleon (NN) interaction. The rearrangement term, indicated by the Hugenholtz–van Hove theorem for the single-particle energy in nuclear matter, is properly considered in the DFM calculation. To validate the use of the density-dependent NN interaction at low energies, an adiabatic approximation was suggested for the dinuclear overlap density. The reliability of the nucleus-nucleus potential predicted through this low-energy version of the DFM was tested in the optical model (OM) analysis of the elastic 12 C+ 12 C and 16 O+ 16 O scattering data at energies below 10 MeV/nucleon. These OM results provide a consistently good description of the elastic angular distributions and 90° excitation function. The dinuclear mean-field potential predicted by the DFM is further used to determine the astrophysical S factor of the 12 C+ 12 C and 16 O+ 16 O fusions in the barrier penetration model. Without any adjustment of the potential strength, our results reproduce the non-resonant behavior of the S factor of the 12 C+ 12 C and 16 O+ 16 O fusions very well over a wide range of energies. Folding model Elastic scattering Fusion 1 Introduction 1 The 12 C+ 12 C and 16 O+ 16 O fusions at low energies, known as the carbon- and oxygen-burning processes, play a vital role in stellar nucleosynthesis [ 1 1 , 2 2 ]. In massive stars with <math id="M1"><mrow><mi>M</mi><mo>≳</mo><mn>8</mn><msub><mi>M</mi><mo>⊙</mo></msub></mrow></math> where the electron degeneracy pressure is insufficient to prevent the gravitational collapse, a large accumulation of 12 C and 16 O ashes built up from the helium-burning phase start to fuse together. In particular, the 12 C+ 12 C fusion occurs at a temperature of (0.6 - 1.0)× 10 9 K and matter density of approximately 10 6 g/cm 3 , and yields 23 Na, 20 Ne, and 23 Mg for further burning stages during the stellar evolution. With a higher Coulomb barrier, the 16 O+ 16 O fusion requires a temperature of (1.5-2.7)× 10 9 K and densities of approximately (2.6-6.7)× 10 9 g/cm 3 to synthesize 28 Si, 31 P, and 31 S. Owing to the important role of the 12 C+ 12 C and 16 O+ 16 O fusions in nuclear astrophysics, numerous studies have been conducted to investigate the fusion cross-sections of these systems at low energies that are close to those relevant for the stellar nucleosynthesis [ 3 3 - 25 25 3-25 ]. In the hot stellar environment, the 12 C+ 12 C and 16 O+ 16 O fusions occur at energies well below the Coulomb barriers of these systems, i.e., around the Gamow peak of approximately 1.5 and 6.6 MeV for the 12 C+ 12 C and 16 O+ 16 O systems, respectively. At such sub-barrier energies, the fusion cross- section falls off exponentially, which makes the direct measurement of the fusion extremely difficult. Therefore, for the astrophysical simulations of the stellar nucleosynthesis, one usually needs to extrapolate the 12 C+ 12 C and 16 O+ 16 O fusion cross-section to the low-energy region based on the experimental data measured at higher energies. However, uncertainties in such a procedure remain very significant [ 10 10 ] owing to the observed resonant structure of the fusion cross-section [ 8 8 , 11 11 , 21 21 ] as well as the discrepancy between the data sets obtained from different measurements in the same energy range, such as discrepancy between the data of the 16 O+ 16 O reaction obtained from the γ -ray and charged-particle detection techniques [ 13 13 , 14 14 ]. Thus, it is of significant interest to have a reliable theoretical prediction of the 12 C+ 12 C and 16 O+ 16 O fusion cross-sections at the astrophysical energies. The theoretical studies to describe the fusion reactions so far are based mainly on the barrier penetration model (BPM) [ 26 26 - 28 28 26-28 ] using different models for the nucleus-nucleus potential [ 18 18 , 19 19 , 21 21 - 25 25 21-25 ]. The results of these studies show that the BPM description of the heavy-ion (HI) fusion depends strongly on the choice of the nucleus-nucleus interaction potential. In general, the validity of a potential model for the description of the 12 C+ 12 C or 16 O+ 16 O interaction at the sub-Coulomb energies should first be tested in a consistent optical model (OM) analysis of elastic scattering at low energies. In fact, the measurements of the 12 C+ 12 C and 16 O+ 16 O elastic scattering and reaction have been performed over a wide range of energies during the last forty years. However, it is not a simple task because of the ambiguity of the optical potential (OP) often observed in the OM studies of the elastic light HI scattering at low energies. In this regard, we remind the readers of the mean-field nature of the light HI interaction [ 29 29 , 30 30 ]. For example, it was shown [ 30 30 ] that the mean-field potential built up from the 12 C+ 12 C interaction smoothly matches the deep family of the real OP that provides a consistently good OM description of the elastic 12 C+ 12 C scattering from the medium energies down to those near the Coulomb barrier. On the Hartree–Fock (HF) level, such a mean-field potential is readily obtained in the double-folding model (DFM) [ 29 29 , 31 31 - 37 37 31-37 ]. At medium energies, the real OP given by the DFM was proven to account very well for the nuclear rainbow pattern observed in elastic light HI scattering [ 29 29 , 35 35 , 37 37 ]. In a smooth extrapolation of the mean-field potential to the low-energy region, the deep double-folded potential was shown [ 30 30 ] to provide a sufficient number of nodes in the relative-motion wave function, as indicated by the Pauli principle, and also provide a natural explanation of the low-energy 12 C+ 12 C resonances in the cluster model of 24 Mg. This indicates that the DFM can be used as a realistic potential model for the description of the 12 C+ 12 C and 16 O+ 16 O fusions at the astrophysical energies. However, during the HI collision at such low energies, the nuclear medium is formed more or less adiabatically, and the nucleus-nucleus overlap density must be treated properly for the DFM calculation using a density-dependent nucleon-nucleon (NN) interaction. This may be the reason that simple versions of the DFM using some density-independent NN interaction are often used to calculate the nucleus-nucleus potential for the fusion study [ 7 7 , 22 22 ]. Instead of the frozen density approximation (FDA) widely used in the DFM calculations at energies above 10 MeV/nucleon [ 34 34 , 35 35 , 37 37 ], we propose an adiabatic density approximation (ADA) for the dinuclear overlap density, which is used in the new version of the DFM [ 37 37 ], which properly includes the rearrangement term of the nuclear mean-field. The present (low-energy) version of the DFM is thereafter carefully tested in the OM analysis of elastic 12 C+ 12 C and 16 O+ 16 O scattering over a wide range of energies below 10 MeV/nucleon. The mean-field potential predicted by the DFM for both the 12 C+ 12 C and 16 O+ 16 O systems is further used in the BPM to determine the (non-resonant) energy-dependent astrophysical S factor for these two systems. 2 Mean-field-based DFM 1 We recall briefly that the nucleus-nucleus OP is evaluated in the DFM as an HF-type potential [ 31 31 , 33 33 - 35 35 33-35 ] using some effective (energy- and density- dependent) NN interactions v ( ρ , E ). The direct term of the double-folded potential is local, and evaluated at a given internuclear distance R using the ground-state (g.s.) densities of the two colliding nuclei as <math display="block" id="M2"><mrow><msub><mi>V</mi><mtext>D</mtext></msub><mo stretchy="false">(</mo><mi>E</mi><mn>,</mn><mi>R</mi><mo stretchy="false">)</mo><mo>=</mo><mstyle displaystyle="true"><mrow><mo>∫</mo></mrow></mstyle><msub><mi>ρ</mi><mi>a</mi></msub><mo stretchy="false">(</mo><msub><mstyle mathvariant="bold" mathsize="normal"><mi>r</mi></mstyle><mi>a</mi></msub><mo stretchy="false">)</mo><msub><mi>ρ</mi><mi>A</mi></msub><mo stretchy="false">(</mo><msub><mstyle mathvariant="bold" mathsize="normal"><mi>r</mi></mstyle><mi>A</mi></msub><mo stretchy="false">)</mo><msub><mi>v</mi><mtext>D</mtext></msub><mo stretchy="false">(</mo><mi>ρ</mi><mn>,</mn><mi>E</mi><mn>,</mn><mi>s</mi><mo stretchy="false">)</mo><msup><mtext>d</mtext><mn>3</mn></msup><msub><mi>r</mi><mi>a</mi></msub><msup><mtext>d</mtext><mn>3</mn></msup><msub><mi>r</mi><mi>A</mi></msub><mn>,</mn><mtext> </mtext><mstyle mathvariant="bold" mathsize="normal"><mi>s</mi></mstyle><mo>=</mo><msub><mstyle mathvariant="bold" mathsize="normal"><mi>r</mi></mstyle><mi>A</mi></msub><mo>−</mo><msub><mstyle mathvariant="bold" mathsize="normal"><mi>r</mi></mstyle><mi>a</mi></msub><mo>+</mo><mstyle mathvariant="bold" mathsize="normal"><mi>R</mi></mstyle><mn>.</mn></mrow></math> The antisymmetrization of the dinuclear system leads to the exchange term V EX that is nonlocal in the coordinate space. A local Wentzel–-Kramers-–Brillouin (WKB)-based approximation is usually made [ 35 35 ] to obtain the exchange term in the following local form: <math display="block" id="M3"><mtable columnalign="left"><mtr><mtd><msub><mi>V</mi><mrow><mtext>E</mtext><mi>X</mi></mrow></msub><mo stretchy="false">(</mo><mi>E</mi><mn>,</mn><mi>R</mi><mo stretchy="false">)</mo><mo>=</mo><mstyle displaystyle="true"><mrow><mo>∫</mo></mrow></mstyle><msub><mi>ρ</mi><mi>a</mi></msub><mo stretchy="false">(</mo><msub><mstyle mathvariant="bold" mathsize="normal"><mi>r</mi></mstyle><mi>a</mi></msub><mn>,</mn><msub><mstyle mathvariant="bold" mathsize="normal"><mi>r</mi></mstyle><mi>a</mi></msub><mo>+</mo><mstyle mathvariant="bold" mathsize="normal"><mi>s</mi></mstyle><mo stretchy="false">)</mo><msub><mi>ρ</mi><mi>A</mi></msub><mo stretchy="false">(</mo><msub><mstyle mathvariant="bold" mathsize="normal"><mi>r</mi></mstyle><mi>A</mi></msub><mn>,</mn><msub><mstyle mathvariant="bold" mathsize="normal"><mi>r</mi></mstyle><mi>A</mi></msub><mo>−</mo><mstyle mathvariant="bold" mathsize="normal"><mi>s</mi></mstyle><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mtext>                </mtext><mo>×</mo><msub><mi>v</mi><mrow><mtext>E</mtext><mi>X</mi></mrow></msub><mo stretchy="false">(</mo><mi>ρ</mi><mn>,</mn><mi>E</mi><mn>,</mn><mi>s</mi><mo stretchy="false">)</mo><mi>exp</mi><mrow><mo>(</mo><mrow><mfrac><mrow><mi>i</mi><mstyle mathvariant="bold" mathsize="normal"><mi>K</mi></mstyle><mo stretchy="false">(</mo><mi>E</mi><mn>,</mn><mi>R</mi><mo stretchy="false">)</mo><mn>.</mn><mstyle mathvariant="bold" mathsize="normal"><mi>s</mi></mstyle></mrow><mi>M</mi></mfrac></mrow><mo>)</mo></mrow><msup><mtext>d</mtext><mn>3</mn></msup><msub><mi>r</mi><mi>a</mi></msub><msup><mtext>d</mtext><mn>3</mn></msup><msub><mi>r</mi><mi>A</mi></msub><mn>,</mn></mtd></mtr></mtable></math> where ρ a ( A ) ( r , r’ ) are the nonlocal g.s. density matrices; M = aA /( a + A ) is the recoil factor (or reduced mass number); a and A are the mass numbers of the projectile and target, respectively. The local relative momentum K ( E , R ) is determined self-consistently as <math display="block" id="M4"><mrow><msup><mi>K</mi><mn>2</mn></msup><mo stretchy="false">(</mo><mi>E</mi><mn>,</mn><mi>R</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mn>2</mn><mi>μ</mi></mrow><mrow><msup><mi>ℏ</mi><mn>2</mn></msup></mrow></mfrac><mo stretchy="false">[</mo><mi>E</mi><mo>−</mo><mi>V</mi><mo stretchy="false">(</mo><mi>E</mi><mn>,</mn><mi>R</mi><mo stretchy="false">)</mo><mo>−</mo><msub><mi>V</mi><mtext>C</mtext></msub><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mn>,</mn></mrow></math> where μ is the reduced mass of the two nuclei and V ( E , R ) = V D ( E , R ) + V EX ( E , R ) is the total real OP. The Coulomb potential V C ( R ) is obtained by directly folding two uniform charge distributions [ 38 38 ], chosen to have the RMS charge radii R C =3.17 and 3.54 fm for 12 C and 16 O, respectively. For the effective NN interaction, in the present work, we have used the CDM3Yn density-dependent version [ 33 33 ] of the M3Y interaction based on the G-matrix elements of Paris potential [ 39 39 ] <math display="block" id="M5"><mrow><msub><mi>v</mi><mrow><mtext>D</mtext><mo stretchy="false">(</mo><mtext>EX</mtext><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>ρ</mi><mn>,</mn><mi>E</mi><mn>,</mn><mi>s</mi><mo stretchy="false">)</mo><mo>=</mo><mi>g</mi><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo><mi>F</mi><mo stretchy="false">(</mo><mi>ρ</mi><mo stretchy="false">)</mo><msub><mi>v</mi><mrow><mtext>D</mtext><mo stretchy="false">(</mo><mtext>EX</mtext><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mn>.</mn></mrow></math> The radial parts of the direct and exchange terms v D(EX) ( s ) were maintained unchanged as given in terms of three Yukawas of the M3Y-Paris interaction [ 39 39 ]. The density-dependent functional F ( ρ ) in Eq. (4) was first suggested in Ref. [ 33 33 ], with parameters chosen to properly reproduce the saturation properties of cold nuclear matter (NM) in the HF calculation (see Fig. 1 Fig. 1 ). The g ( E ) factor accounts effectively for the (in-medium) energy dependence of the CDM3Yn interaction (4), and is determined self-consistently [ 37 37 ] using the local relative momentum (3). In the HF calculation of the single-nucleon energy in NM using a density-dependent NN interaction, there appears naturally a rearrangement term (RT), which accounts for the rearrangement of the mean-field caused by the removal or addition of a single nucleon. The significant impact of the RT was shown experimentally in the direct nucleon transfer reactions at low energies [ 40 40 ]. In the same mean-field manner, the RT must appear in the HF-type folding model calculation of the nucleon-nucleus or nucleus-nucleus potential using an explicitly density-dependent NN interaction and single-nucleon wave functions of the projectile and target nucleons. This important aspect of the folding model has been investigated recently [ 37 37 , 41 41 ], and it was shown that the contribution of the RT to the double-folded potential (2)-(3) can be accurately accounted for by adding a density-dependent correction term Δ F ( ρ ) to the density dependence F ( ρ ) of the CDM3Yn interaction, <math display="block" id="M6"><mrow><msub><mi>v</mi><mrow><mtext>D</mtext><mo stretchy="false">(</mo><mtext>EX</mtext><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>ρ</mi><mn>,</mn><mi>E</mi><mn>,</mn><mi>s</mi><mo stretchy="false">)</mo><mo>=</mo><mi>g</mi><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo><mo stretchy="false">[</mo><mi>F</mi><mo stretchy="false">(</mo><mi>ρ</mi><mo stretchy="false">)</mo><mo>+</mo><mi>Δ</mi><mi>F</mi><mo stretchy="false">(</mo><mi>ρ</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><msub><mi>v</mi><mrow><mtext>D</mtext><mo stretchy="false">(</mo><mtext>EX</mtext><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></mrow></math> must be used in the DFM calculation (2)-(3) of the nucleus-nucleus OP. The explicit parameters of the CDM3Yn interaction (4) with n = 3 and 6, and the corresponding RT correction (5) are given in Ref. [ 37 37 ]. In our approach, the contribution of the RT is added to the total double-folded potential as <math display="block" id="M7"><mrow><mi>V</mi><mo stretchy="false">(</mo><mi>E</mi><mn>,</mn><mi>R</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>V</mi><mrow><mtext>H</mtext><mi>F</mi></mrow></msub><mo stretchy="false">(</mo><mi>E</mi><mn>,</mn><mi>R</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mi>V</mi><mrow><mtext>R</mtext><mi>T</mi></mrow></msub><mo stretchy="false">(</mo><mi>E</mi><mn>,</mn><mi>R</mi><mo stretchy="false">)</mo><mn>,</mn></mrow></math> where V HF and V RT are the HF-type and rearrangement terms of the double-folded potential, respectively. The g.s. densities of 12 C and 16 O given by the no-core shell model [ 42 42 ] and Hartree–Fock–Bogoliubov (HFB) calculations [ 43 43 ], respectively, were used in the present work. Adiabatic density approximation 2 As the strength and shape of the double-folded potential at small radii depend strongly on the dinuclear overlap density at these distances [ 35 35 ], an appropriate treatment of the overlap density is very important for the DFM calculation using a density-dependent NN interaction. The FDA, widely used so far in the double-folding calculation, assumes the sum of the two "frozen" g.s. densities for the overlap density ρ in Eq. (5). <math display="block" id="M8"><mrow><mi>ρ</mi><mo>=</mo><msub><mi>ρ</mi><mn>1</mn></msub><mo stretchy="false">(</mo><msub><mstyle mathvariant="bold" mathsize="normal"><mi>r</mi></mstyle><mn>1</mn></msub><mo stretchy="false">)</mo><mo>+</mo><msub><mi>ρ</mi><mn>2</mn></msub><mo stretchy="false">(</mo><msub><mstyle mathvariant="bold" mathsize="normal"><mi>r</mi></mstyle><mn>2</mn></msub><mo stretchy="false">)</mo><mn>.</mn></mrow></math> The FDA was proven to be a reliable approximation at energies above 10 MeV/nucleon [ 31 31 - 33 33 31-33 , 35 35 , 36 36 ]. At low energies of the sub-barrier fusion, the dinuclear system merges adiabatically into the compound nucleus [ 44 44 - 47 47 44-47 ] at decreasing internuclear distance R , and the FDA for a frozen overlap density is no longer valid. In an adiabatic scenario, the fusion time at low energies is sufficiently long for nucleons in the compound system to rearrange their initial states while maintaining the total energy of the system as low as possible. The ADA proposed here is similar to that suggested earlier in Refs. [ 48 48 , 49 49 ], and the dinuclear overlap density evolves with the internuclear separation R as <math display="block" id="M9"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>ρ</mi><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mo>{</mo><mrow><mtable><mtr><mtd><mrow><mn>0.5</mn><msub><mi>ρ</mi><mtext>c</mtext></msub><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo></mrow></mtd><mtd><mrow><mtext>exp</mtext><mrow><mo>[</mo><mrow><mtext>ln</mtext><mrow><mo>(</mo><mrow><mfrac><mrow><msub><mi>ρ</mi><mn>0</mn></msub><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo></mrow><mrow><mn>0.5</mn><msub><mi>ρ</mi><mtext>c</mtext></msub><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mfrac><mi>R</mi><mrow><msub><mi>R</mi><mrow><mtext>cut</mtext></mrow></msub></mrow></mfrac></mrow><mo>)</mo></mrow></mrow><mo>]</mo></mrow><mtext> </mtext></mrow></mtd></mtr><mtr><mtd><mrow/></mtd><mtd><mrow><mi>i</mi><mi>f</mi><mtext> </mtext><mtext> </mtext><mi>R</mi><mo>⩽</mo><msub><mi>R</mi><mrow><mtext>cut</mtext></mrow></msub></mrow></mtd></mtr><mtr><mtd><mrow><msub><mi>ρ</mi><mn>0</mn></msub><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo></mrow></mtd><mtd><mrow><mtext> </mtext><mi>i</mi><mi>f</mi><mtext> </mtext><mtext> </mtext><mi>R</mi><mo>></mo><msub><mi>R</mi><mrow><mtext>cut</mtext></mrow></msub><mn>,</mn></mrow></mtd></mtr></mtable></mrow></mrow></mrow></mtd></mtr></mtable></mrow></math> where R cut is the grazing distance at which the two colliding nuclei start to rearrange themselves to form the compound nucleus; ρ 0 ( r ) and ρ c ( r ) are the g.s. densities of 12 C (or 16 O) nucleus and of the compound 24 Mg (or 32 S) nucleus, respectively. The 12 C+ 12 C overlap densities given by the FDA and ADA are illustrated in Fig. 2 Fig. 2 . The chosen R cut = 6 fm is close to the grazing distance suggested from the coupled channels study of fusion at sub-Coulomb energies [ 47 47 ]. It can be observed that the overlap density given by the FDA increases strongly at short distances, to approximately twice the central density of 12 C at R =0, whereas that given by the ADA approaches the density of 24 Mg nucleus at small radii. The impact of the two different treatments of the overlap density on the strength and shape of the double-folded 12 C+ 12 C potential at the energy of approximately 7 MeV/nucleon obtained with the CDM3Y3 interaction is illustrated Fig. 3 Fig. 3 . As the overlap density given by the ADA is softer than that given by the FDA (see Fig. 2 Fig. 2 ), the corresponding ADA double-folded potential is much deeper and has a larger slope in comparison with the FDA potential in the interior ( R < 6 fm). Such a difference in the shape of the double-folded potential can be observed in the OM results for the elastic 12 C+ 12 C scattering shown in Fig. 4 Fig. 4 . 3 Results and discussion 1 3.1 Elastic 12 C+ 12 C scattering and astrophysical S factor 2 The DFM should first be tested to verify its reliability in the OM study of elastic 12 C+ 12 C scattering at low energies before using it as the potential model in the BPM calculation of the fusion cross-section. For this purpose, the OM analysis of the elastic 12 C+ 12 C scattering data at energies below 10 MeV/nucleon was performed using the CDM3Y3 double-folded potential as the real OP and Woods–Saxon (WS) potential as the imaginary OP, so that the total OP at the internuclear distance R is determined as <math display="block" id="M10"><mrow><mi>U</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo><mo>=</mo><mi>V</mi><mo stretchy="false">(</mo><mi>E</mi><mn>,</mn><mi>R</mi><mo stretchy="false">)</mo><mo>+</mo><mi>i</mi><msub><mi>W</mi><mi>V</mi></msub><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo><mo>+</mo><mi>i</mi><msub><mi>W</mi><mi>D</mi></msub><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mi>V</mi><mtext>C</mtext></msub><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo><mn>,</mn></mrow></math> <math display="block" id="M11"><mrow><mtext>with</mtext><mtext> </mtext><msub><mi>W</mi><mi>V</mi></msub><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mo>−</mo><msub><mi>W</mi><mi>V</mi></msub></mrow><mrow><mn>1</mn><mo>+</mo><mi>exp</mi><mo stretchy="false">[</mo><mo stretchy="false">(</mo><mi>R</mi><mo>−</mo><msub><mi>R</mi><mi>V</mi></msub><mo stretchy="false">)</mo><mo>/</mo><msub><mi>a</mi><mi>V</mi></msub><mo stretchy="false">]</mo></mrow></mfrac></mrow></math> <math display="block" id="M12"><mrow><mtext>and</mtext><mtext> </mtext><msub><mi>W</mi><mi>D</mi></msub><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo><mo>=</mo><mn>4</mn><msub><mi>a</mi><mi>D</mi></msub><mfrac><mtext>d</mtext><mrow><mtext>d</mtext><mi>R</mi></mrow></mfrac><mrow><mo>{</mo><mrow><mfrac><mrow><msub><mi>W</mi><mi>D</mi></msub></mrow><mrow><mn>1</mn><mo>+</mo><mi>exp</mi><mo stretchy="false">[</mo><mo stretchy="false">(</mo><mi>R</mi><mo>−</mo><msub><mi>R</mi><mi>D</mi></msub><mo stretchy="false">)</mo><mo>/</mo><msub><mi>a</mi><mi>D</mi></msub><mo stretchy="false">]</mo></mrow></mfrac></mrow><mo>}</mo></mrow><mn>.</mn></mrow></math> The WS surface term (11) is optional and is used only in the case of the elastic 16 O+ 16 O scattering. The WS parameters in (11) were obtained from the global OP of the 12 C+ 12 C system [ 29 29 , 50 50 ], with the potential strength slightly adjusted by the OM fit to the data under study (see Table 1 Table 1 ). The Coulomb potential V C ( R ) is obtained by directly folding two uniform charge distributions [ 37 37 ] chosen to have the RMS charge radii R C =3.17 and 3.54 fm for 12 C and 16 O ions, respectively. All the OM calculations were performed using the code ECIS97 written by Raynal [ 51 51 ]. 1 1 System 1 1 E lab (MeV) 1 1 W V (MeV) 1 1 R V (fm) 1 1 a V (fm) 1 1 W D (MeV) 1 1 R D (fm) 1 1 a D (fm) 1 1 12 C+ 12 C 1 1 83.3 1 1 8.672 1 1 6.367 1 1 0.387 1 1 - 1 1 - 1 1 - 1 1   1 1 102.1 1 1 14.326 1 1 5.584 1 1 0.591 1 1 - 1 1 - 1 1 - 1 1   1 1 117.1 1 1 15.770 1 1 5.536 1 1 0.590 1 1 - 1 1 - 1 1 - 1 1 16 O+ 16 O 1 1 48.5 1 1 8.254 1 1 5.351 1 1 0.282 1 1 0.708 1 1 7.823 1 1 0.221 1 1   1 1 53.0 1 1 9.424 1 1 5.353 1 1 0.231 1 1 0.571 1 1 7.818 1 1 0.243 1 1   1 1 63.0 1 1 14.170 1 1 5.373 1 1 0.380 1 1 3.428 1 1 6.997 1 1 0.354 The OM results for the elastic 12 C+ 12 C scattering at 83, 102, and 117 MeV given by the real CDM3Y3 double-folded OP using the FDA and ADA for the overlap density are compared with the measured data in Fig. 4 Fig. 4 . It can be observed that the folded potential obtained with the overlap density given by the FDA fails to account for the data measured at large angles, which are sensitive to the real OP at small distances [ 35 35 ] where the two approximations for the overlap density result in different potential strengths as shown in Fig. 3 Fig. 3 . Upon using a more realistic ADA for the overlap density, the double-folded potential provides a good description of the data at both forward and backward angles without any adjustment of the potential strength. The boson symmetry of the identical 12 C+ 12 C system leads to the Mott oscillation of elastic cross-section at large angles (see Fig. 4 Fig. 4 ), which is peaked at θ c.m. =90°. Therefore, the elastic excitation function has been measured at 90∘ for this system at energies ranging from the Coulomb barrier to above 10 MeV/nucleon. A complex structure of the peaks and valleys in the measured 12 C+ 12 C excitation function (see Fig. 5 Fig. 5 )was a challenge during the eighties, which was solved by McVoy and Brandan in a mean-field study of elastic 12 C+ 12 C scattering [ 50 50 ] where they showed that the uneven structure in the 90∘ 12 C+ 12 C excitation function is due to the evolution of the rainbow (Airy) pattern. In particular, the prominent minimum at 102 MeV is caused by the second Airy minimum passing through 90∘ at that energy [ 50 50 ]. As shown in Fig. 4 Fig. 4 , the elastic data measured at 102 MeV and two neighboring energies can be reproduced only by the real double-folded ADA potential. These two versions of the real OP and the WS imaginary parts extrapolated from the global OP for elastic 12 C+ 12 C scattering at the same energy range [ 29 29 , 50 50 ] were used to calculate the 90∘ excitation function. We observed that only the ADA potential can reproduce the measured excitation function over a wide range of energies (see Fig. 5 Fig. 5 ). Therefore, we conclude that the present version of the DFM provides a reliable prediction of the real OP for the 12 C+ 12 C system at low energies and it should be a suitable potential model for the BPM study of the 12 C+ 12 C fusion at the astrophysical energies. In the BPM, the probability of the 12 C+ 12 C fusion is determined by the tunnel effect that allows two 12 C nuclei to penetrate the Coulomb barrier at energies below the barrier height. Given the Q value of nearly 14 MeV, the 12 C+ 12 C fusion can proceed through different channels involving the compound 24 Mg nucleus. While it is complicated to consider all these reaction channels properly in the coupled reaction channel calculation, the total 12 C+ 12 C fusion cross-section can be determined in the BPM as a coherent sum of all partial-wave contributions of the 12 C+ 12 C transmission <math display="block" id="M13"><mrow><msub><mi>σ</mi><mrow><mtext>fus</mtext></mrow></msub><mo>=</mo><mfrac><mi>π</mi><mrow><msup><mi>k</mi><mn>2</mn></msup></mrow></mfrac><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>l</mi><mo>=</mo><mn>0</mn></mrow><mi>∞</mi></munderover></mstyle><mrow><mo>[</mo><mrow><mn>1</mn><mo>+</mo><msup><mrow><mo stretchy="false">(</mo><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><mi>l</mi></msup></mrow><mo>]</mo></mrow><mo stretchy="false">(</mo><mn>2</mn><mi>l</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><msub><mi>T</mi><mi>l</mi></msub><mn>,</mn></mrow></math> where k is the relative-motion momentum and l is the orbital angular momentum of the dinuclear system. The transmission coefficient T l provides the probability of the two nuclei tunneling through the potential barrier built up from the attractive nuclear potential and repulsive Coulomb and centrifugal potentials as <math display="block" id="M14"><mrow><msub><mi>V</mi><mi>l</mi></msub><mo stretchy="false">(</mo><mi>R</mi><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><msub><mi>V</mi><mtext>C</mtext></msub><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo><mo>+</mo><mfrac><mrow><msup><mi>ℏ</mi><mn>2</mn></msup><mi>l</mi><mo stretchy="false">(</mo><mi>l</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><mrow><mn>2</mn><mi>μ</mi><msup><mi>R</mi><mn>2</mn></msup></mrow></mfrac><mn>.</mn></mrow></math> The double-folded potential based on the ADA for the overlap density is used as the nuclear potential V N . For consistency, the same folded Coulomb potential V C as that obtained in Eq. (3) is used to determine the total potential (13). To illustrate the impact of the potential model on the BPM results, we have also used two other models of the nuclear potential in the BPM study of the 12 C+ 12 C fusion. The first one is the WS parameterization, denoted hereafter as the BRA potential, suggested for the OP of the 12 C+ 12 C system at energies below 6 MeV/nucleon [ 55 55 ]. The other is the empirical potential parameterized to properly describe the g.s. band of 24 Mg in the 12 C+ 12 C cluster model [ 56 56 ], denoted hereafter as the BH potential. In general, the probability of fusion depends strongly on the height and location of the potential barrier. The double-folded (based on the ADA for the overlap density) potential provides the barrier height V B0 = 6.2 MeV at R B0 = 7.8 fm, which are consistent with the empirical values deduced [ 5 5 ] from the measured fusion data, V B0 ≈ 6.2-6.3 MeV and R B0 ≈ 7.4-7.6 fm. It can be observed in Fig. 6 Fig. 6 that the double-folded potential provides a good description of the 12 C+ 12 C fusion cross-section over the entire energy range. The barrier heights and positions predicted by the BRA potential ( V B0 =5.5 MeV and R B0 =8.4 fm) and the BH potential ( V B0 =6.4 MeV and R B0 =7.2 fm) differ significantly from the empirical values. Consequently, the BPM results given by the BRA potential overestimate the fusion data in the energy region below 7 MeV whereas those given by the BH potential underestimate the data (see Fig. 6 Fig. 6 ). As the fusion cross-section decreases exponentially with the decrease in energy, it is more convenient to consider the astrophysical S -factor of the fusion <math display="block" id="M15"><mrow><mi>S</mi><mo>=</mo><mi>E</mi><msub><mi>σ</mi><mrow><mtext>fus</mtext></mrow></msub><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo><mi>exp</mi><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><mi>η</mi><mo stretchy="false">)</mo><mn>,</mn></mrow></math> where the Sommerfeld parameter η = Z 1 Z 2 e 2 / ħv , and v is the relative velocity of the dinuclear system. The astrophysical S factors obtained in the BPM with the three potential models are compared in Fig. 7 Fig. 7 with the data, and the empirical extrapolation by Fowler et al. [ 2 2 ] based on the data measured at higher energies. It can be observed that both the BRA and BH potentials fail to reproduce the data in the same way as discussed for the fusion cross-section. The double-folded (based on the ADA for the overlap density) potential provides a good BPM description of the non-resonant behavior of the S factor over the entire energy range. The BPM results given by the double-folded potential are also reasonably consistent with the phenomenological S factor suggested by Fowler et al. , except at the lowest energies where our mean-field results are slightly lower. 3.2 Results obtained for the 16 O+ 16 O system 2 Given the higher Coulomb barrier, the 16 O+ 16 O fusion occurs at higher temperature and densities compared with those of the stellar condition for the 12 C+ 12 C fusion. So far, the S factor measured for the 16 O+ 16 O fusion seems to be smooth, in contrast to that of the 12 C+ 12 C fusion, which has uneven resonant behavior at low energies. Therefore, it is of interest to extend our mean-field approach to study the 16 O+ 16 O fusion at energies of astrophysical interest. For this purpose, the 16 O+ 16 O potential has been calculated in the DFM using the CDM3Y3 interaction and ADA for the overlap density, and the g.s. densities of 16 O and 32 S given by the Green Function Monte Carlo [ 57 57 ] and HFB [ 43 43 ] calculations, respectively. In the OM analysis, this version of DFM was used as the real OP whereas the imaginary OP was assumed in the WS form with parameters given in Table 1 Table 1 . Without any renormalization of the potential strength, the OM results given by the double-folded potential account very well for the elastic 16 O+ 16 O scattering data at low energies (see Fig. 8 Fig. 8 ). In the BPM study of the 16 O+ 16 O fusion at energies around the Coulomb barrier, the double-folded potential (based on the ADA for the overlap density) provides a fairly good description of the measured S factor over a wide energy range, as shown in Fig. 9 Fig. 9 . The results given by the double-folded potential are also reasonably consistent with the phenomenological S factor suggested by Fowler et al. [ 2 2 ]. As in the 12 C+ 12 C case, we have compared our mean-field results with those given by another potential model for the 16 O+ 16 O fusion. That is, the phenomenological potential suggested by Wu et al. [ 15 15 ] for the study of the elastic 16 O+ 16 O scattering at low energies and fusion has been used in the present BPM calculation. It can be observed in Fig. 9 Fig. 9 that the phenomenological potential suggested by Wu et al. significantly underestimates the data at sub-barrier energies. The results shown in Fig. 9 Fig. 9 confirm the reliability of the low-energy version of the DFM in predicting the nucleus-nucleus potential for the BPM study of nuclear fusion. The S factor given by the double-folded potential seems to slightly underestimate the data at low energies. A dynamic contribution from the coupled channel effects might well enhance the sub-barrier fusion cross-section. This will be the subject of our future study. 4 Summary 1 The mean-field based CDM3Y3 density-dependent interaction [ 33 33 ], well tested in the HF studies of nuclear matter [ 35 35 ] and nucleon mean-field potential [ 37 37 , 41 41 ], has been used in the DFM approach to calculate the real OP for the elastic 12 C+ 12 C and 16 O+ 16 O scattering at low energies. With a realistic adiabatic approximation for the overlap density, the double-folded potential accounts very well for the observed oscillation and magnitude of the elastic 12 C+ 12 C and 16 O+ 16 O cross-sections over a wide angular angle. The present version of the DFM is further used to calculate the nuclear mean-field potential for the study of the astrophysical S factor of the 12 C+ 12 C and 16 O+ 16 O fusion in the BPM. Without any free parameter, the double-folded potential accounts very well for the non-resonant strength of the 12 C+ 12 C and 16 O+ 16 O astrophysical S factors. Our results are also consistent with those given by the phenomenological S factor extrapolated by Fowler et al. [ 2 2 ] from the data measured for these two systems. The OP given by the present version of the DFM can be used as a reliable input in the coupled reaction channels study of the 12 C+ 12 C and 16 O+ 16 O fusion, to explore explicitly the dynamic contribution from the coupled reaction channels to the total fusion cross-section and astrophysical S factor at sub-barrier energies. References C. Iliadis , Nuclear Physics of Stars , Wiley-VCH Press , Weinheim ( 2015 ). Nuclear Physics of Stars W.A. Fowler , G.R. Caughlan , B.A. Zimmerman , Thermonuclear Reaction Rates, II . Ann. Rev. Astron. Astrophys . 13 , 69 - 112 ( 1975 ). doi: 10.1146/annurev.aa.13.090175.000441 http://doi.org/10.1146/annurev.aa.13.090175.000441 Thermonuclear Reaction Rates, II K.U. Kettner , H. Lorenz-Wirzba , C. Rolfs , The 12C+12C reaction at subcoulomb energies (I) . Z. Phys. A 298 , 65 - 75 ( 1980 ). doi: 10.1007/BF01416030 http://doi.org/10.1007/BF01416030 The 12C+12C reaction at subcoulomb energies (I) W. Treu , H. Fröhlich , W. Galster et al ., Total reaction cross section for 12C+12C in the vicinity of the Coulomb barrier . Phys. Rev. C 22 , 2462 - 2464 ( 1980 ). doi: 10.1103/PhysRevC.22.2462 http://doi.org/10.1103/PhysRevC.22.2462 Total reaction cross section for 12C+12C in the vicinity of the Coulomb barrier E.F. Aguilera , P. Rosales , E. Martinez-Quiroz et al ., New γ-ray measurements for 12C+12C 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 L. Barrón-Palos , E.F. Aguilera , J. Aspiazu et al ., Absolute cross sections measurement for the 12C+12C system at astrophysically relevant energies . Nucl. Phys. A 779 , 318 - 332 ( 2006 ). doi: 10.1016/j.nuclphysa.2006.09.004 http://doi.org/10.1016/j.nuclphysa.2006.09.004 Absolute cross sections measurement for the 12C+12C system at astrophysically relevant energies L.R. Gasques , A.V. Afanasjev , E.F. Aguilera et al ., Nuclear fusion in dense matter: Reaction rate and carbon burning . Phys. Rev. C 72 , 025806 ( 2005 ). doi: 10.1103/PhysRevC.72.025806 http://doi.org/10.1103/PhysRevC.72.025806 Nuclear fusion in dense matter: Reaction rate and carbon burning T. Spillane , F. Raiola , C. Rolfs et al ., 12C+12C fusion reactions near the Gamow energy . Phys. Rev. Lett . 98 , 122501 ( 2007 ). doi: 10.1103/PhysRevLett.98.122501 http://doi.org/10.1103/PhysRevLett.98.122501 12C+12C fusion reactions near the Gamow energy B. Bucher , X.D. Tang , X. Fang et al ., First direct measurement of 12C(12C,n)23Mg at stellar energies . Phys. Rev. Lett . 114 , 251102 ( 2015 ). doi: 10.1103/PhysRevLett.114.251102 http://doi.org/10.1103/PhysRevLett.114.251102 First direct measurement of 12C(12C,n)23Mg at stellar 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 A. Tumino , C. Spitaleri , M. La Cognata et al ., An increase in the 12C+12C fusion rate from resonances at astrophysical energies . Nature 557 , 687 - 690 ( 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 A. Diaz-Torres , M. Wiescher , Characterizing the astrophysical S factor for 12C+12C fusion with wave-packet dynamics . Phys. Rev. C 97 , 055802 ( 2018 ). doi: 10.1103/PhysRevC.97.055802 http://doi.org/10.1103/PhysRevC.97.055802 Characterizing the astrophysical S factor for 12C+12C fusion with wave-packet dynamics H. Spinka , H. Winkler , Experimental determination of the total reaction cross section of the stellar nuclear reaction 16O+16O . Nucl. Phys. A 233 , 456 - 494 ( 1974 ). doi: 10.1016/0375-9474(74)90469-2 http://doi.org/10.1016/0375-9474(74)90469-2 Experimental determination of the total reaction cross section of the stellar nuclear reaction 16O+16O G. Hulke , C. Rolfs , H.P. Trautvetter , Comparison of the fusion reactions 12C+20Ne and 16O+16O near the Coulomb barrier . Z. Phys. A 297 , 161 - 183 ( 1980 ). doi: 10.1007/BF01421473 http://doi.org/10.1007/BF01421473 Comparison of the fusion reactions 12C+20Ne and 16O+16O near the Coulomb barrier S.C. Wu , C.A. Barnes , Fusion and elastic scattering cross sections for the 16O+16O reactions near the Coulomb barrier . Nucl. Phys. A 422 , 373 - 396 ( 1984 ). doi: 10.1016/0375-9474(84)90523-2 http://doi.org/10.1016/0375-9474(84)90523-2 Fusion and elastic scattering cross sections for the 16O+16O reactions near the Coulomb barrier J.G. Duarte , L.R. Gasques , J.R.B. Oliveira et al ., Measurement of fusion cross sections for 16O+16O . J. Phys. G: Nucl. Part. Phys . 42 , 065102 ( 2015 ). doi: 10.1088/0954-3899/42/6/065102 http://doi.org/10.1088/0954-3899/42/6/065102 Measurement of fusion cross sections for 16O+16O C.L. Jiang , K.E. Rehm , B.B. Back , R.V.F. Janssens , Expectations for 12C and 16O induced fusion cross sections at energies of astrophysical interest . Phys. Rev. C 75 , 015803 ( 2007 ). doi: 10.1103/PhysRevC.75.015803 http://doi.org/10.1103/PhysRevC.75.015803 Expectations for 12C and 16O induced fusion cross sections at energies of astrophysical interest V.Yu. Denisov , N.A. Pilipenko , Fusion of deformed nuclei: 12C+12C . Phys. Rev. C 81 , 025805 ( 2010 ). doi: 10.1103/PhysRevC.81.025805 http://doi.org/10.1103/PhysRevC.81.025805 Fusion of deformed nuclei: 12C+12C M. Notani , H. Esbensen , X. Fang et al ., Correlation between the 12C+12C, 12C+13C, and 13C+13C fusion cross sections . Phys. Rev. C 85 , 014607 ( 2012 ). doi: 10.1103/PhysRevC.85.014607 http://doi.org/10.1103/PhysRevC.85.014607 Correlation between the 12C+12C, 12C+13C, and 13C+13C fusion cross sections C.L. Jiang , B.B. Back , H. Esbensen et al ., Origin and consequences of 12C+12C 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 M. Assunção , P. Descouvemont , Role of the Hoyle state in 12C+12C fusion . Phys. Lett. B 723 , 355 - 359 ( 2013 ). doi: 10.1016/j.physletb.2013.05.030 http://doi.org/10.1016/j.physletb.2013.05.030 Role of the Hoyle state in 12C+12C fusion A.A. Aziz , N. Yusof , M.Z. Firihu , H.A. Kassim , Reliability of the double-folding potential for fusion cross sections of light systems . Phys. Rev. C 91 , 015811 ( 2015 ). doi: 10.1103/PhysRevC.91.015811 http://doi.org/10.1103/PhysRevC.91.015811 Reliability of the double-folding potential for fusion cross sections of light systems H. Esbensen , Coupled-channels calculations of 16O+16O fusion . Phys. Rev. C 77 , 054608 ( 2008 ). doi: 10.1103/PhysRevC.77.054608 http://doi.org/10.1103/PhysRevC.77.054608 Coupled-channels calculations of 16O+16O fusion G. Kocak , M. Karakoc , I. Boztosun , A.B. Balantekin , Effects of α-cluster potentials for the 16O+16O fusion reaction and S factor . Phys. Rev. C 81 , 024615 ( 2010 ). doi: 10.1103/PhysRevC.81.024615 http://doi.org/10.1103/PhysRevC.81.024615 Effects of α-cluster potentials for the 16O+16O fusion reaction and S factor C. Simenel , R. Keser , A.S. Umar , V.E. Oberacker , Microscopic study of 16O+16O fusion . Phys. Rev. C 88 , 024617 ( 2013 ). doi: 10.1103/PhysRevC.88.024617 http://doi.org/10.1103/PhysRevC.88.024617 Microscopic study of 16O+16O fusion C.Y. Wong , Interaction barrier in charged-particle nuclear reactions . Phys. Rev. Lett . 31 , 766 - 769 ( 1973 ). doi: 10.1103/PhysRevLett.31.766 http://doi.org/10.1103/PhysRevLett.31.766 Interaction barrier in charged-particle nuclear reactions K. Hagino , N. Takigawa , Subbarrier fusion reactions and many-particle quantum tunneling . Prog. Theor. Phys . 128 , 1061 - 1106 ( 2012 ). doi: 10.1143/PTP.128.1061 http://doi.org/10.1143/PTP.128.1061 Subbarrier fusion reactions and many-particle quantum tunneling B.B. Back , H. Esbensen , C.L. Jiang , K.E. Rehm , Recent developments in heavy-ion fusion reactions . Rev. Mod. Phys . 86 , 317 - 360 ( 2014 ). doi: 10.1103/RevModPhys.86.317 http://doi.org/10.1103/RevModPhys.86.317 Recent developments in heavy-ion fusion reactions M.E. Brandan , G.R. Satchler , The interaction between light heavy-ions and what it tells us . Phys. Rep . 285 , 143 - 243 ( 1997 ). doi: 10.1016/S0370-1573(96)00048-8 http://doi.org/10.1016/S0370-1573(96)00048-8 The interaction between light heavy-ions and what it tells us Y. Kondō , M.E. Brandan , G.R. Satchler , Shape resonances and deep optical potentials: A mean-field description of 12C+12C scattering at low energies . Nucl. Phys. A 637 , 175 - 200 ( 1998 ). doi: 10.1016/S0375-9474(98)00212-7 http://doi.org/10.1016/S0375-9474(98)00212-7 Shape resonances and deep optical potentials: A mean-field description of 12C+12C scattering at low energies G.R. Satchler , W.G. Love , Folding model potentials from realistic interactions for heavy-ion scattering . Phys. Rep . 55 , 183 - 254 ( 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 , W. von Oertzen , H.G. Bohlen , Double-folding model for heavy-ion optical potential: Revised and applied to study 12C and 16O elastic scattering . Phys. Rev. C 49 , 1652 - 1668 ( 1994 ). doi: 10.1103/PhysRevC.49.1652 http://doi.org/10.1103/PhysRevC.49.1652 Double-folding model for heavy-ion optical potential: Revised and applied to study 12C and 16O elastic scattering D.T. Khoa , G.R. Satchler , W. von Oertzen , Nuclear incompressibility and density dependent NN interactions in the folding model for nucleus-nucleus potentials . Phys. Rev. C 56 , 954 - 969 ( 1997 ). doi: 10.1103/PhysRevC.56.954 http://doi.org/10.1103/PhysRevC.56.954 Nuclear incompressibility and density dependent NN interactions in the folding model for nucleus-nucleus potentials D.T. Khoa , G.R. Satchler , Generalized folding model for elastic and inelastic nucleus-nucleus scattering using realistic density dependent nucleon-nucleon interaction . Nucl. Phys. A 668 , 3 - 41 ( 2000 ). doi: 10.1016/S0375-9474(99)00680-6 http://doi.org/10.1016/S0375-9474(99)00680-6 Generalized folding model for elastic and inelastic nucleus-nucleus scattering using realistic density dependent nucleon-nucleon interaction D.T. Khoa , W. von Oertzen , H.G. Bohlen , S. Ohkubo , Nuclear rainbow scattering and nucleus-nucleus potential . J. Phys. G 34 , R111 - R164 ( 2007 ). doi: 10.1088/0954-3899/34/3/R01 http://doi.org/10.1088/0954-3899/34/3/R01 Nuclear rainbow scattering and nucleus-nucleus potential 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 D.T. Khoa , N.H. Phuc , D.T. Loan , B.M. Loc , Nuclear mean field and double-folding model of the nucleus-nucleus optical potential . Phys. Rev. C 94 , 034612 ( 2016 ). doi: 10.1103/PhysRevC.94.034612 http://doi.org/10.1103/PhysRevC.94.034612 Nuclear mean field and double-folding model of the nucleus-nucleus optical potential J.E. Poling , E. Norbeck , R.R. Carlson , Elastic scattering of lithium by 9Be, 10B, 12C, 13C, 16O, and 28Si from 4 to 63 MeV . Phys. Rev. C 13 , 648 - 660 ( 1976 ). doi: 10.1103/PhysRevC.13.648 http://doi.org/10.1103/PhysRevC.13.648 Elastic scattering of lithium by 9Be, 10B, 12C, 13C, 16O, and 28Si from 4 to 63 MeV N. Anantaraman , H. Toki , G.F. Bertsch , An effective interaction for inelastic scattering derived from the Paris potential . Nucl. Phys. A 398 , 269 - 278 ( 1983 ). doi: 10.1016/0375-9474(83)90487-6 http://doi.org/10.1016/0375-9474(83)90487-6 An effective interaction for inelastic scattering derived from the Paris potential P.E. Hodgson , Nucleon removal and rearrangement energies . Rep. Prog. Phys . 38 , 847 - 902 ( 1975 ). doi: 10.1088/0034-4885/38/7/002 http://doi.org/10.1088/0034-4885/38/7/002 Nucleon removal and rearrangement energies D.T. Loan , B.M. Loc , D.T. Khoa , Extended Hartree-Fock study of the single-particle potential: The nuclear symmetry energy, nucleon effective mass, and folding model of the nucleon optical potential . Phys. Rev. C 92 , 034304 ( 2015 ). doi: 10.1103/PhysRevC.92.034304 http://doi.org/10.1103/PhysRevC.92.034304 Extended Hartree-Fock study of the single-particle potential: The nuclear symmetry energy, nucleon effective mass, and folding model of the nucleon optical potential M. Gennari , M. Vorabbi , A. Calci , P. Navrátil , Microscopic optical potentials derived from ab initio translationally invariant nonlocal one-body densities . Phys. Rev. C 97 , 034619 ( 2018 ). doi: 10.1103/PhysRevC.97.034619 http://doi.org/10.1103/PhysRevC.97.034619 Microscopic optical potentials derived from ab initio translationally invariant nonlocal one-body densities S. Goriely , M. Samyn , J.M. Pearson , Further explorations of Skyrme-Hartree-Fock-Bogoliubov mass formulas . VII. Simultaneous fits to masses and fission barriers. Phys. Rev. C 75 , 064312 ( 2007 ). doi: 10.1103/PhysRevC.75.064312 http://doi.org/10.1103/PhysRevC.75.064312 Further explorations of Skyrme-Hartree-Fock-Bogoliubov mass formulas K. Siwek-Wilczýnska , J. Wilczýnski , Nucleus-nucleus fusion energy thresholds and the adiabatic fusion potential . Phys. Rev. C 64 , 024611 ( 2001 ). doi: 10.1103/PhysRevC.64.024611 http://doi.org/10.1103/PhysRevC.64.024611 Nucleus-nucleus fusion energy thresholds and the adiabatic fusion potential 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 Y. Taniguchi , Y. Kanada-Enýo , T. Suhara , Adiabatic internuclear potentials obtained by energy variation with the internuclear-distance constraint . Prog. Theor. Exp. Phys . 2013 , 043D04 ( 2013 ). doi: 10.1093/ptep/ptt002 http://doi.org/10.1093/ptep/ptt002 Adiabatic internuclear potentials obtained by energy variation with the internuclear-distance constraint 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 I. Reichstein , F.B. Malik , Dependence of 16O+16O potential on the density ansatz . Phys. Lett. B 37 , 344 - 346 ( 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 N. Ohtsuka , R. Linden , A. Faessler , F.B. Malik , Real and imaginary parts of the microscopic optical potential between nuclei in the sudden and adiabatic approximation and its application to medium energy 12C+12C scattering . Nucl. Phys. A 465 , 550 - 572 ( 1987 ). doi: 10.1016/0375-9474(87)90364-2 http://doi.org/10.1016/0375-9474(87)90364-2 Real and imaginary parts of the microscopic optical potential between nuclei in the sudden and adiabatic approximation and its application to medium energy 12C+12C scattering K.W. McVoy , M.E. Brandan , The 900 excitation function for elastic 12C+12C scattering: The importance of Airy elephants . Nucl. Phys. A 542 , 295 - 309 ( 1992 ). doi: 10.1016/0375-9474(92)90218-9 http://doi.org/10.1016/0375-9474(92)90218-9 The 900 excitation function for elastic 12C+12C scattering: The importance of Airy elephants J. Raynal , Computing as a Language of Physics , IAEA , Vienna p. 75 ( 1972 ); Computing as a Language of Physics coupled-channel code ECIS97 (unpublished) . coupled-channel code ECIS97 (unpublished) R.G. Stokstad , R.M. Wieland , G.R. Satchler et al ., Elastic and inelastic scattering of 12C by 12C from Ec.m. = 35-63 MeV . Phys. Rev. C 20 , 655 - 669 ( 1979 ). doi: 10.1103/PhysRevC.20.655 http://doi.org/10.1103/PhysRevC.20.655 Elastic and inelastic scattering of 12C by 12C from Ec.m. = 35-63 MeV A. Morsad , F. Haas , C. Beck , R.M. Freeman , Detailed energy dependence of the elastic and inelastic 12C+12C scattering between 5 and 10 MeV/nucleon . Z. Phys. A 338 , 61 - 65 ( 1991 ). doi: 10.1007/BF01279115 http://doi.org/10.1007/BF01279115 Detailed energy dependence of the elastic and inelastic 12C+12C scattering between 5 and 10 MeV/nucleon W. Reilly , R. Wieland , A. Gobbi et al ., A comparative study of the elastic scattering in the 12C,14N and 16O identical-particle systems . Nuov. Cim. A 13 , 897 - 912 ( 1973 ). doi: 10.1007/BF02804157 http://doi.org/10.1007/BF02804157 A comparative study of the elastic scattering in the 12C,14N and 16O identical-particle systems M.E. Brandan , M. Rodrguez-Villafuerte , A. Ayala , 12C+12C elastic scattering analysis above E/A=6 MeV using deep real potentials . Phys. Rev. C 41 , 1520 - 1529 ( 1990 ). doi: 10.1103/PhysRevC.41.1520 http://doi.org/10.1103/PhysRevC.41.1520 12C+12C elastic scattering analysis above E/A=6 MeV using deep real potentials B. Buck , P.D.B. Hopkins , A.C. Merchant , A 12C+12C cluster model of 24Mg . Nucl. Phys. A 513 , 75 - 114 ( 1990 ). doi: 10.1016/0375-9474(90)90344-L http://doi.org/10.1016/0375-9474(90)90344-L A 12C+12C cluster model of 24Mg S.C. Pieper , R.B. Wiringa , V.R. Pandharipande , Variational calculation of the ground state of 16O . Phys. Rev. C 46 , 1741 - 1756 ( 1992 ). doi: 10.1103/PhysRevC.46.1741 http://doi.org/10.1103/PhysRevC.46.1741 Variational calculation of the ground state of 16O R.H. Siemssen , J.V. Maher , A. Weidinger , D.A. Bromley , Excitation-Function Structure in 16O+16O Scattering . Phys. Rev. Lett . 19 , 369 - 372 ( 1967 ). doi: 10.1103/PhysRevLett.19.369 http://doi.org/10.1103/PhysRevLett.19.369 Excitation-Function Structure in 16O+16O Scattering J.V. Maher , M.W. Sachs , R.H. Siemssen et al ., Nuclear Interaction of Oxygen with Oxygen . Phys. Rev . 188 , 1665 - 1683 ( 1969 ). doi: 10.1103/PhysRev.188.1665 http://doi.org/10.1103/PhysRev.188.1665 Nuclear Interaction of Oxygen with Oxygen 10.1007/s41365-018-0517-7 The present research has been supported, in part, by the National Foundation for Scientific and Technological Development (NAFOSTED Project No. 103.04-2017.317). 2018-09-11 2018-10-18 2018-10-31 2018-11-26 2018-12-01 2021-12-20 © 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. 2018 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/s41365-018-0517-7 http://dx.doi.org/10.1007/s41365-018-0517-7 . 2018 Nuclear Science and Techniques 12 29