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Register a new user$mathbf{{}^{12}{C} + {}^{12}{C}}$ Fusion $boldsymbol{S^*}$-factor from a Full-microscopic Nuclear Model
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The ${}^{12}mathrm{C} + {}^{12}mathrm{C}$ fusion reaction plays a vital role in the explosive phenomena of the universe. The resonances in the Gamow window rule its reaction rate and products. Hence, the determination of the resonance parameters by nuclear models is indispensable as the direct measurement is not feasible. Here, for the first time, we report the resonances in the ${}^{12}mathrm{C} + {}^{12}mathrm{C}$ fusion reaction described by a full-microscopic nuclear model. The model plausibly reproduces the measured low-energy astrophysical $S$-factors and predicts the resonances in the Gamow window. Contradictory to the hindrance model, we conclude that there is no low-energy suppression of the $S$-factor.
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A quantitative study of the astrophysically important sub-barrier fusion of $^{12}$C+$^{12}$C is presented. Low-energy collisions are described in the body-fixed reference frame using wave-packet dynamics within a nuclear molecular picture. A collective Hamiltonian drives the time propagation of the wave-packet through the collective potential-energy landscape. The fusion imaginary potential for specific dinuclear configurations is crucial for understanding the appearance of resonances in the fusion cross section. The theoretical sub-barrier fusion cross sections explain some observed resonant structures in the astrophysical S-factor. These cross sections monotonically decline towards stellar energies. The structures in the data that are not explained are possibly due to cluster effects in the nuclear molecule, which are to be included in the present approach.
The $^{12}$C+$^{12}$C fusion reaction plays a crucial role in stellar evolution and explosions. Its open reaction channels mainly include $alpha$, $p$, $n$, and ${}^{8}$Be. Despite more than a half century of efforts, large discrepancies remain among the experimental data measured using various techniques. In this work, we analyze the existing data using the statistical model. Our calculation shows: 1) the relative systematic uncertainties of the predicted branching ratios get smaller as the predicted ratios increase; 2) the total modified astrophysical S-factors (S$^*$ factors) of the $p$ and $alpha$ channels can each be obtained by summing the S$^*$ factors of their corresponding ground-state transitions and the characteristic $gamma$ rays while taking into account the contributions of the missing channels to the latter. After applying corrections based on branching ratios predicted by the statistical model, an agreement is achieved among the different data sets at ${E}_{cm}>$4 MeV, while some discrepancies remain at lower energies suggesting the need for better measurements in the near future. We find that the recent S$^*$ factor obtained from an indirect measurement is inconsistent with the direct measurement at energies below 2.6 MeV. We recommend upper and lower limits for the ${}^{12}$C+${}^{12}$C S$^*$ factor based on the existing models. A new $^{12}$C+$^{12}$C reaction rate is also recommended.
Lowest energy spectrum of the $^{12}$C nucleus is analyzed in the 3$alpha$ cluster model with a deep $alphaalpha$-potential of Buck, Friedrich and Wheatley with Pauli forbidden states in the $S$ and $D$ waves. The direct orthogonalization method is applied for the elimination of the 3$alpha$-Pauli forbidden states. The effects of possible first order quantum phase transition are shown in the lowest $^{12}$C($0_1^+)$ and $^{12}$C($2_1^+)$ states from weakly bound phase to a deep phase. The ground and lowest $2^+$ states of the $^{12}$C nucleus in the deep phase are created by the critical eigen states of the Pauli projector for the $0^+$ and $2^+$ three-alpha functional spaces, respectively.
Form factors for $alpha+{^{12}}$C inelastic scattering are obtained within two theoretical ($alpha+alpha+alpha$) approaches: The hyperspherical framework for three identical bosons, and the algebraic cluster model assuming the $D_{3h}$ symmetry of an equilateral triangle subject to rotations and vibrations. Results show a good agreement, with form factors involving the Hoyle state having a slightly larger extension within the hyperspherical approach. Coupled-channel calculations using these form factors are ongoing.
We use an underground counting lab with an extremely low background to perform an activity measurement for the $^{12}$C+$^{13}$C system with energies down to $Erm_{c.m.}$=2.323 MeV, at which the $^{12}$C($^{13}$C,$p$)$^{24}$Na cross section is found to be 0.22(7) nb. The $^{12}$C+$^{13}$C fusion cross section is derived with a statistical model calibrated using experimental data. Our new result of the $^{12}$C+$^{13}$C fusion cross section is the first decisive evidence in the carbon isotope systems which rules out the existence of the astrophysical S-factor maximum predicted by the phenomenological hindrance model, while confirming the rising trend of the S-factor towards lower energies predicted by other models, such as CC-M3Y+Rep, DC-TDHF, KNS, SPP and ESW. After normalizing the model predictions with our data, a more reliable upper limit is established for the $^{12}$C+$^{12}$C fusion cross sections at stellar energies.
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