Spatial measure of reaction size in proton scattering

The Hoyle state in $^{12}$C has a developed 3$\alpha $ cluster structure, and its matter radius is expected to be enhanced by about 50 percent in comparison to the radius of the ground state. However, the enhanced radius of the 3$\alpha $ state is not confirmed experimentally. Recently we have proposed ``the scattering radius,'' which characterizes a spatial size of an exclusive reaction in a general two-body scattering problem. In the present study, we perform the microscopic coupled-channel calculation for the proton $+ ^{12}$C system, and the scattering radii for the inelastic scatterings to various excited states are evaluated. The proton - $^{12}$C nuclear interactions are derived from the folding model, which employs the density-dependent M3Y effective nucleon-nucleon interaction and the $^{12}$C transition densities, obtained from the microscopic 3$\alpha $ cluster model. We have calculated the angular distributions for the inelastic scattering to the collective states (2$_{1}^{+}$ and 3$_{1}^{-}$) and the 3$\alpha $ cluster states (0$_{2}^{+}$ and 2$_{2}^{+}$). The scattering radii are derived for the individual channels, and we have confirmed the strong enhancement of the scattering radii in the 3$\alpha $ channels, which is consistent to the picture of the nuclear $\alpha $ condensation. In the present report, we will explain the enhancement of the scattering radii in the 3$\alpha $ channels in connection to the matter radii of the 3$\alpha $ cluster states.