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Register a new userPeculiarity of the $^{12}$C$(0^+)$ and $^{12}$C$(2^+)$ energy spectrum in a 3$alpha$ model
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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.
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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.
Geometric configurations of three-$alpha$ particles in the ground- and first-excited $J^pi=0^+$ states of $^{12}$C are discussed within two types of $alpha$-cluster models which treat the Pauli principle differently. Though there are some quantitative differences especially in the internal region of the wave functions, equilateral triangle configurations are dominant in the ground state, while in the first excited $0^+$ state isosceles triangle configurations dominate, originating from $^8{rm Be}+alpha$ configurations.
Densities and transition densities are computed in an equilateral triangular alpha-cluster model for $^{12}$C, in which each $alpha$ particle is taken as a gaussian density distribution. The ground-state, the symmetric vibration (Hoyle state) and the asymmetric bend vibration are analyzed in a molecular approach and dissected into their components in a series of harmonic functions, revealing their intrinsic structures. The transition densities in the laboratory frame are then used to construct form-factors and to compute DWBA inelastic cross-sections for the $^{12}$C$(alpha, alpha)$ reaction. The comparison with experimental data indicates that the simple geometrical model with rotations and vibrations gives a reliable description of reactions where $alpha$-cluster degrees of freedom are involved.
The molecular algebraic model based on three and four alpha clusters is used to describe the inelastic scattering of alpha particles populating low-lying states in $^{12}$C and $^{16}$O. Optical potentials and inelastic formfactors are obtained by folding densities and transition densities obtained within the molecular model. One-step and multi-step processes can be included in the reaction mechanism calculation. In spite of the simplicity of the approach the molecular model with rotations and vibrations provides a reliable description of reactions where $alpha$-cluster degrees of freedom are involved and good results are obtained for the excitation of several low-lying states. Within the same model we briefly discuss the expected selection rules for the $alpha$-transfer reactions from $^{12}$C and $^{16}$O.
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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