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Register a new userThe algebraic molecular model in $^{12}$C and its application to the $alpha$+$^{12}$C scattering: from densities and transition densities to optical potentials and nuclear formfactors
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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.
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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.
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.
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.
Elastic $^{16}$O+$^{12}$C scattering is known to exhibit the nuclear rainbow pattern at incident energies $E_text{lab}gtrsim 200$ MeV, with the Airy structure of the far-side scattering cross section clearly seen at medium and large angles. Such a rainbow pattern is well described by the deep real optical potential (OP) given by the double-folding model (DFM). At lower energies, the extensive elastic $^{16}$O+$^{12}$C scattering data show consistently that the nuclear rainbow pattern at backward angles is deteriorated by an oscillating enhancement of elastic cross section that is difficult to describe in the conventional optical model (OM). Given a significant $alpha$ spectroscopic factor predicted for the dissociation $^{16}$O$toalpha+^{12}$C by the shell model and $alpha$-cluster models, the contribution of the elastic $alpha$ transfer (or the core-core exchange) to the elastic $^{16}$O+$^{12}$C scattering should not be negligible and is expected to account for the enhanced elastic cross section at backward angles. To reveal the impact of the elastic $alpha$ transfer, a systematic coupled reaction channels analysis of the elastic $^{16}$O+$^{12}$C scattering has been performed, with the coupling between the elastic scattering and elastic $alpha$ transfer channels treated explicitly, using the real OP given by the DFM. We found that the elastic $alpha$ transfer enhances the near-side scattering significantly at backward angles, giving rise to an oscillating distortion of the smooth Airy structure. The dynamic polarization of the OP by the coupling between the elastic scattering and elastic $alpha$ transfer channels can be effectively taken into account in the OM calculation by an angular-momentum (or parity) dependent potential added to the imaginary OP, as suggested by Frahn and Hussein 40 years ago.
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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