We propose a method for determining the nuclear radii for the excited states lying above the particle breakup threshold based on the diffraction model of scattering. The method is applied to analyzing the diffraction structure of the elastic and inelastic scattering of ${}^2\mathrm{He}$, ${}^3\mathrm{He}$, ${}^4\mathrm{He}$, ${}^6\mathrm{Li}$, and ${}^{12}\mathrm{C}$ ions on ${}^{12}\mathrm{C}$ at energies below 100 MeV/$A$. We study the radii of ${}^{12}\mathrm{C}$ in the excited states up to $E_x\approx 11$ MeV and show that the diffraction radii for the ground and the first $2^{+}$ ($4.44$ MeV) excited states are approximately the same. The diffraction radii for the $0_2^{+}$ ($7.65$ MeV) Hoyle state and $3^{-}$ ($9.64$ MeV) states located above the ${}^{12}\mathrm{C}$ $\to 3\alpha$ threshold are larger by $\approx 0.5$ fm. This difference does not depend on the energy or on the kind of projectiles (deuterons are an exception). This fact justifies an application of the proposed method to the determination of the root-mean-square radii of the above-threshold states. We found that the rms radii for the $0_2^{+}$ ($7.65$ MeV) and $3^{-}$ ($9.64$ MeV) states are a factor of $1.2$ larger than the rms radius for the ground state of ${}^{12}\mathrm{C}$. Also, we estimated the rms radii for the above-threshold $2^{+}$ ($9.9$ MeV), $0^{+}$ ($10.3$ MeV), and $1^{-}$ ($10.84$ MeV) states.

• Received 13 August 2009

DOI:https://doi.org/10.1103/PhysRevC.80.054603