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Register a new userThree-body description of $boldsymbol{^{12}}$C: From the hyperspherical formulation to the algebraic cluster model and its application to $boldsymbol{alpha}+boldsymbol{^{12}}$C inelastic scattering
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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.
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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.
The algebraic molecular model is used in $^{12}$C to construct densities and transition densities connecting low-lying states of the rotovibrational spectrum, first and foremost those belonging to the rotational bands based on the ground and the Hoyle states. These densities are then used as basic ingredients to calculate, besides electromagnetic transition probabilities, nuclear potentials and formfactors to describe elastic and inelastic $alpha$+$^{12}$C scattering processes. The calculated densities and transition densities are also compared with those obtained by directly solving the problem of three interacting alphas within a three-body approach where continuum effects, relevant in particular for the Hoyle state, are properly taken into account.
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.
We carry out an ab initio calculation of the neutrino flux-folded inclusive cross sections, measured on $^{12}$C by the MiniBooNE and T2K collaborations in the charged-current quasielastic (CCQE) regime. The calculation is based on realistic two- and three-nucleon interactions, and on a realistic nuclear electroweak current with one-and two-nucleon terms that are constructed consistently with these interactions and reproduce low-energy electroweak transitions. Numerically exact quantum Monte Carlo methods are utilized to compute the nuclear weak response functions, by fully retaining many-body correlations in the initial and final states and interference effects between one- and two-body current contributions. We employ a nucleon axial form factor of the dipole form with $Lambda_A = 1.0$ or $1.15$ GeV, the latter more in line with a very recent lattice QCD determination. The calculated cross sections are found to be in good agreement with the neutrino data of MiniBooNE and T2K, and antineutrino MiniBooNE data, yielding a consistent picture of nuclei and their electroweak properties across a wide regime of energy and momenta.
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.
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