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$\alpha$ structure. A prominent enhancement of the scattering radius is clearly confirmed in the 3$\alpha$ 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$\alpha$ 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.

Figure 1

Incident-energy dependence of the mean radius of the imaginary potentials. The solid squares show the radii for the 0${}_2^{+}$ channel, while the crosses are the radii for the ground 0${}_1^{+}$ channel.

Figure 2

Differential cross section at $E_p=65$ MeV. Panel (a) shows the result of the elastic (0${}_1^{+}$) channel, while the result of the inelastic (0${}_2^{+}$) channel is plotted in panel (b). In both panels, the asterisks and curves represent the experimental data and the theoretical calculations, respectively.

Figure 3

Distributions of partial cross sections at $E_p=65$ MeV. The asterisks represent the partial cross sections $\sigma \left(J\right)$ for the elastic scattering, while the distribution with the squares shows the cross section for the inelastic scattering. The inelastic partial cross sections are multiplied by a factor of 200 to be shown in the same scale as the elastic scattering.

Figure 4

Partial cross sections for inelastic scattering at $E_p=65$ MeV. The squares show the results of the full CC calculation, while the circles show the results without any potentials in the exit 0${}_2^{+}$ channel. The distribution with the squares is the same as the distribution with the squares shown in Fig. 3.

Figure 5

Radial shape of the coupling [$V_{\text{cp}}\left(R\right)$] and diagonal [$V_{\text{dg}}\left(R\right)$] potentials. The diagonal potential multiplied by $R^2$ for the elastic 0${}_1^{+}$ channel is shown by the dashed curve, while that for the inelastic 0${}_2^{+}$ channel is shown by the thick solid curve. The coupling potential of $0_1^{+}\to 0_2^{+}$ is shown by the dotted curve. The right-side ordinate corresponds to the strength of the diagonal potentials, while the coupling potential is plotted with the left-side ordinate. The magnitude of both diagonal potentials are multiplied by the factor 1$$8. The arrow at $R=4$ fm shows the distance, at which the absorption effect becomes strong. See text for details.

Figure 6

Radial part of the transition density for $0_1^{+}\to 0_2^{+}$. The squared radius $r^2$ is multiplied to the transition density. The solid curve shows the density calculated by the microscopic 3$\alpha$ cluster model, while the dotted curve represents the density obtained by the BM model. The BM density is multiplied by a minus sign.

Figure 7

Comparison of differential cross sections for inelastic scattering at $E_p=65$ MeV. The solid and dotted curves represent the calculation based on the cluster and BM transition densities, respectively.

Figure 8

Distributions of the partial cross section in the inelastic scattering at $E_p=65$ MeV. The asterisks show the distribution for the 3$\alpha$ cluster state, 0${}_2^{+}$, while the squares show that for the GMR state.

Figure 9

Excitation energy dependence of scattering radius. The abscissa represents the excitation energy, while the ordinate represents the scattering radius $R_{\text{SC}}$. The solid squares and crosses denote the $R_{\text{SC}}$ calculated for $E_p=65$ and 200 MeV, respectively. The dotted line at $E_x=7.65$ MeV represents the excitation energy of the 3$\alpha$ cluster excitation, while the line at $E_x=35$ MeV shows the energy of the GMR excitation.

Figure 10

Energy dependence of scattering radius $R_{\text{SC}}$. The solid squares show $R_{\text{SC}}$ for the 3$\alpha$ excitation, while the solid circles show the radius for the elastic scattering. The asterisks represent $R_{\text{SC}}$ for GMR excitation. The upper and lower dotted lines show the matter radius of the 0${}_2^{+}$ and 0${}_1^{+}$ states, respectively. The white symbols are the diffraction radii derived from proton scattering at $E_p=1040$ MeV [11]. See text for details.