Detailed Studies of 12C Structure and Reactions

Abstract

We are reporting here on a series of theoretical investigations with both algebraic models and geometric cluster models of alpha clusters in $^{12}$C, focusing on the structure of the ground state, the first excited $0^+$ state and the second excited $2^+$ state with the purpose, in particular, of establishing if the rotational bands are compatible with rigid structures or rather if they are quantum mixture of different configurations. In a first series of paper (Vitturi et al., Transition densities and form factors in the triangular $\alpha$-cluster model of 12C with application to 12C+$\alpha$ scattering. Phys Rev C 101:014315, 2020; Casal et al., Alpha-induced inelastic scattering and alpha-transfer reactions in 12C and 16O within the Algebraic Cluster Model. Eur Phys J A 57:33, 2021), we assume a rigid equilateral triangle shape and study in detail several properties that descend from the algebraic framework, such as the energy spectrum, electromagnetic observables and calculate the transition densities in order to extract elastic and inelastic cross-sections for various processes. In a second series of papers (Moriya et al., Three-$\alpha$ Configurations in the 0$^+$ States of 12C. Few-Body Syst 62:46, 2021; Moriya et al., Three-$\alpha$ configurations of the second $J^\pi$ = 0$^+$ state in 12C. Eur. Phys J A 59:37, 2023), we solve the three-body Schrödinger equation with orthogonality conditions using the stochastic variational method with correlated Gaussian basis functions. The two-body density distributions indicate that the main configurations of both the $0_2^+$ and $2_2^+$ states are acute iscosceles triangle shapes coming from $^8$Be($0^+$)+$\alpha$ configurations and find some hints that the second $2^+$ state is not an ideal rigid rotational band member of the Hoyle state band.