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Register a new userAlpha-induced inelastic scattering and alpha-transfer reactions in $^{12}$C and $^{16}$O within the Algebraic Cluster Model
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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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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.
The elastic scattering $^{16}$O$+^{12}$C angular distributions at $^{16}$O bombarding energies of 100.0, 115.9 and 124.0 MeV and their optical model description including the $alpha$-particle exchange contribution calculated in the Coupled Reaction Channel approach are presented. The angular distributions show not only the usual diffraction pattern but also, at larger angles, intermediate structure of refractive origin on which finer oscillations are superimposed. The large angle features can be consistently described including explicitly the elastic $alpha$-transfer process and using a refractive optical potential with a deep real part and a weakly absorptive imaginary part.
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.
Inelastic $^{16}$O +$^{12}$C rainbow scattering to the $2^+$ (4.44 MeV) state of $^{12}$C was measured at the incident energies, $E_L$ = 170, 181, 200, 260 and 281 MeV. A systematic analysis of the experimental angular distributions was performed using the coupled channels method with an extended double folding potential derived from realistic wave functions for $^{12}$C and $^{16}$O calculated with a microscopic $alpha$ cluster model and a finite-range density-dependent nucleon-nucleon force.The coupled channels analysis of the measured inelastic scattering data shows consistently some Airy-like structure in the inelastic scattering cross sections for the first $2^+$ state of $^{12}$C, which is somewhat obscured and still not clearly visible in the measured data. The Airy minimum was identified from the analysis and the systematic energy evolution of the Airy structure was studied. The Airy minimum in inelastic scattering is found to be shifted backward compared with that in elastic scattering.
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