Measure of the spatial size for the monopole excitation in proton scattering

We formulate a scattering radius, which will be demonstrated to be a good measure of the spatial size of a general exclusive reaction. The scattering radius is presented in a framework of the partial-wave expansion method in a general two-body scattering problem. A microscopic coupled-channel calculation is performed for proton scattering by $^{12}\mathrm{C}$ in the range of the proton's incident energy, ${E}_{p}=29.95--200$ MeV, and the scattering radii are evaluated for elastic scattering and inelastic scattering, going to the Hoyle 0${}_{2}^{+}$ state with a well-developed 3$\ensuremath{\alpha}$ structure. A prominent enhancement of the scattering radius is clearly confirmed in the 3$\ensuremath{\alpha}$ final channel in comparison to the elastic channel. The scattering radius is also calculated for excitation to the giant monopole resonance (GMR) in a microscopic coupled-channel framework. The scattering radius for the 3$\ensuremath{\alpha}$ excitation is much more enhanced than the scattering radius for the GMR excitation. The proton's incident-energy dependence of the scattering radius is also investigated, and the energy systematics strongly suggest that the scattering radius can characterize the spatial size of a reaction area, which is determined by the matter radius of a nucleus excited to a final state.