The 2⁺ excitation of the Hoyle state

Abstract

To understand why the 2⁺ excitation of the Hoyle state was so difficult to observe in the direct reaction experiments with the ¹²C target, a detailed folding model + coupled-channel analysis of the inelastic α+¹²C scattering at Elab = 240 and 386 MeV has been done using the complex optical potential and inelastic scattering form factor obtained from the double-folding model using the nuclear transition densities predicted by the antisymmetrized molecular dynamics. With the complex strength of the density dependent nucleon-nucleon interaction fixed by the optical model description of the elastic α+¹²C scattering, the inelastic scattering form factor was fine tuned to the best coupled-channel description of the (α, α′) cross section measured for each excited state of ¹²C, and the corresponding isoscalar Eλ transition strength has been accurately determined. The present analysis of the (α, α′) data measured in the energy bins around Ex ≈ 10 MeV has unambiguously revealed the E2 transition strength that should be assigned to the 2₂⁺ state of ¹²C. A very weak transition strength B(E2; 0₁⁺ → 2₂⁺) ≈ 3 e²fm⁴ has been established, which is smaller than the E2 strength predicted for the transition from the Hoyle state to the 2₂⁺ state by at least two orders of magnitude. This is one of the main reasons why the direct excitation of the 2₂⁺ state of ¹²C has been difficult to observe in the experiments.