$\alpha$ scattering cross sections on ${}^{12}\mathrm{C}$ with a microscopic coupled-channels calculation

$\alpha$ elastic and inelastic scattering on ${}^{12}\mathrm{C}$ is investigated with the coupled-channels calculation by using microscopic $\alpha -{}^{12}\mathrm{C}$ potentials, which are derived by folding the Melbourne $g$-matrix $NN$ interaction with the densities of $\alpha$ and ${}^{12}\mathrm{C}$. The matter and transition densities of ${}^{12}\mathrm{C}$ are obtained by a microscopic structure model of the antisymmetrized molecular dynamics combined with and without the $3\alpha$ generator coordinate method. The calculation reproduces satisfactorily well the observed elastic and inelastic cross sections at incident energies of $E_{\alpha }=130$, 172.5, 240, and 386 MeV without phenomenological fitting parameters adjusted to hadron scattering reactions. Isoscalar monopole and dipole excitations to the $0_2^{+}, 0_3^{+}$, and $1_1^{-}$ states in the $\alpha$ scattering are discussed.

Figure 1

Squared charge form factors of ${}^{12}\mathrm{C}$. The theoretical values are those obtained with AMD and $\text{AMD}+\text{GCM}$ scaled by the factor $f_{\text{tr}}^2$ (labeled by ${\mathrm{AMD}}_{f_{\text{tr}}}$ and $\text{AMD}+{\mathrm{GCM}}_{f_{\text{tr}}}$, respectively). The experimental data are those measured by electron elastic and inelastic scattering on ${}^{12}\mathrm{C}$ from Refs. [60, 62, 63, 64].

Figure 2

$\alpha$ scattering cross sections on ${}^{12}\mathrm{C}$ at $E_{\alpha }=130$ MeV ($\times {10}^4$), 172.5 MeV ($\times {10}^2$), 240 MeV, and 386 MeV ($\times {10}^{-2}$). The differential cross sections of the $0_1^{+}, 0_2^{+}, 0_3^{+}, 2_1^{+}, 2_2^{+}$, and $3_1^{-}$ states obtained by the CCs calculation with the AMD and $\text{AMD}+\text{GCM}$ transition densities are shown. The cross sections obtained by the DWBA calculation with the AMD transition densities are also shown for comparison. The experimental data are taken from Refs. [35, 37, 45, 48, 49]. References for those data are summarized in Table 3.

Figure 3

Same as Fig. 2, but for the $1_1^{-}, 1_2^{-}, 4_1^{+}$, and $4_2^{+}$ states.

Figure 4

$2_2^{+}$ and $0_3^{+}$ cross sections at $E_{\alpha }=240$ and 386 MeV calculated with AMD and $\text{AMD}+\text{GCM}$. The incoherent sum of the $2_2^{+}$ and $0_3^{+}$ cross sections at 386 MeV is compared with the experimental sum of the $2_2^{+}\left(9.84\right)$ MeV and $0_3^{+}$ (9.93 MeV) taken from Ref. [35]. The experimental $0_3^{+}$ cross sections at 240 MeV are taken from Ref. [37].

Figure 5

$\alpha$ scattering cross sections on ${}^{12}\mathrm{C}$ at $E_{\alpha }=130$ MeV ($\times {10}^4$), 172.5 MeV ($\times {10}^2$), 240 MeV, and 386 MeV ($\times {10}^{-2}$), obtained by the CCs calculation with the RGM densities compared with the AMD result. The calculated differential cross sections of the $0_1^{+}, 0_2^{+}, 2_1^{+}, 2_2^{+}$, and $3_1^{-}$ states are shown. The experimental data from Refs. [35, 37, 45, 48, 49] are also shown.

Figure 6

Proton densities ${\rho }_p\left(r\right)=\rho \left(r\right)/2$ of the AMD, $\text{AMD}+\text{GCM}$, and RGM calculations.

Figure 7

Transition densities ${\rho }^{\mathrm{tr}}\left(r\right)$ of the AMD, $\text{AMD}+\text{GCM}$, and RGM calculations. The calculated densities scaled with a factor $I_f\equiv \sqrt{2J_f+1}$ are plotted.