Folding-model analysis of inelastic $\ensuremath{\alpha}{+}^{12}$C scattering at medium energies, and the isoscalar transition strengths of the cluster states of ${}^{12}$C

Folding-model analysis of inelastic $α +^{12}$ C scattering at medium energies, and the isoscalar transition strengths of the cluster states of $^{12}$ C

Do Cong Cuong, Dao T. Khoa, and Yoshiko Kanada-En'yo

Phys. Rev. C 88, 064317 – Published 17 December 2013

Abstract

Background: The (spin and isospin zero) $α$ -particle is an efficient projectile for the excitation of the isoscalar, natural-parity states of $^{12}$ C. Among those states that have pronounced $α$ -cluster structure, the Hoyle state ( $0_{2}^{+}$ state at 7.65 MeV) has been observed in many ( $α, α^{'}$ ) experiments while the second $2^{+}$ state of $^{12}$ C, predicted at $E_{x} \approx 10$ MeV as an excitation of the Hoyle state, has not been observed until a recent high-precision experiment of the $α +^{12}$ C scattering at $E_{α} = 386$ MeV. A plausible reason is a strong population of the narrow $3_{1}^{-}$ state at 9.64 MeV and broad 0 $_{3}^{+}$ resonance at 10.3 MeV that hinders the weak $2_{2}^{+}$ peak in the ( $α, α^{'}$ ) spectrum.

Purpose: The accurate determination of the electric $E λ$ transition strengths of the isoscalar states of $^{12}$ C, including an $E 2$ component at $E_{x} \approx 10$ MeV that can be assigned to the $2_{2}^{+}$ state, based on a detailed folding-model + coupled-channels analysis of the ( $α, α^{'}$ ) data measured at $E_{α} = 240$ and 386 MeV.

Method: The complex optical potential and inelastic form factor given by the folding model for the $α +^{12}$ C scattering are used to calculate the ( $α, α^{'}$ ) cross sections for the known isoscalar states of $^{12}$ C in an elaborate coupled-channels approach. The strengths of the form factors for these states are then fine tuned against the ( $α, α^{'}$ ) data to deduce the corresponding $E λ$ transition strengths.

Results: A significant $E 2$ transition strength has been obtained for the $2_{2}^{+}$ state from the present analysis of the ( $α, α^{'}$ ) data measured at $E_{α} = 240$ and 386 MeV. The $E λ$ transition strengths of the $0_{2}^{+}, 3_{1}^{-}, 0_{3}^{+},$ and $1_{1}^{-}$ states were also carefully deduced, and some difference from the results of earlier analyses has been found and qualitatively understood.

Conclusion: Despite a strong hindrance by the $3_{1}^{-}$ and $0_{3}^{+}$ excitations, the presence of the $2_{2}^{+}$ state in the ( $α, α^{'}$ ) spectra measured at $E_{α} = 240$ and 386 MeV has been consistently confirmed by the present folding-model + coupled-channels analysis.