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.

我们在这里报告一系列关于\(^{12}\) C 中 α 簇的代数模型和几何簇模型的理论研究，重点关注基态的结构，第一激发态\(0^+\ )态和第二激发态\(2^+\)，其目的特别是确定旋转能带是否与刚性结构兼容，或者它们是否是不同配置的量子混合。在第一个系列论文中（Vitturi 等人，12C 三角\(\alpha \)簇模型中的过渡密度和形状因子及其应用于 12C+ \(\alpha \)散射。Phys Rev C 101:014315， 2020；Casal 等人，Algebraic Cluster Model 中的 12C 和 16O 中的 Alpha-induced inelasticscattering and alpha-transfer reports.Eur Phys JA 57:33, 2021)，我们假设一个刚性的等边三角形形状并详细研究了几个属性源自代数框架，例如能谱、电磁可观测量，并计算转变密度，以便提取各种过程的弹性和非弹性横截面。在第二系列论文中（Moriya et al., Three- \(\alpha \) Configurations in the 0 \(^+\) States of 12C. Few-Body Syst 62:46, 2021；Moriya et al., Three - 12C 中第二个\(J^\pi \) = 0 \(^+\)态的 \ (\alpha \)配置. Eur. Phys JA 59:37, 2023)，我们求解三体薛定谔方程使用具有相关高斯基函数的随机变分方法的正交性条件。二体密度分布表明\(0_2^+\)和\(2_2^+\)态的主要构型都是锐角等腰三角形，来自\(^8\) Be( \(0^+ \) )+ \(\alpha \)配置并发现一些提示，表明第二个\(2^+\)态不是霍伊尔态带的理想刚性旋转带成员。