First microscopic coupled-channels calculation of cross sections for inelastic $\alpha$ scattering off ${}^{16}\mathrm{O}$

The $\alpha$ inelastic scattering on ${}^{16}\mathrm{O}$ is investigated with the coupled-channels calculation using the $\alpha$-nucleus coupled-channels potentials, which are microscopically derived by folding the the Melbourne $g$-matrix $NN$ interaction with the ${}^{16}\mathrm{O}$ and $\alpha$ densities. The matter and transition densities of ${}^{16}\mathrm{O}$ are calculated by a microscopic structure model of the variation after the spin-parity projections combined with the generator coordinate method of ${}^{12}\mathrm{C}+\alpha$ in the framework of the antisymmetrized molecular dynamics. The calculation reproduces the observed elastic and inelastic cross sections at incident energies of $E_{\alpha }=104$, 130, 146, and 386 MeV. The coupled-channels effect on the cross sections is also discussed.

Figure 1

Proton density ${\rho }_p\left(r\right)=\rho \left(r\right)/2$ of the $0_{1,2,3,4,5}^{+},1_{1,2}^{-},2_{1,2,3}^{+},3_{1,2}^{-}$, and $4_{1,2}^{+}$ states of ${}^{16}\mathrm{O}$ calculated with the $\mathrm{VAP}+\mathrm{GCM}$.

Figure 2

The energy levels adopted in the CC calculation and the $E2$ transitions. The positive-parity states ($0^{+},2^{+}$, and $4^{+}$) are shown by open triangles and the negative-parity states ($1^{-}$ and $3^{-}$) are shown by open squares. The energy levels are connected by (green) dashed and (red) solid lines in the cases of $50<B\left(E2\right)<100e^2{\mathrm{fm}}^4$ and $B\left(E2\right)>100e^2{\mathrm{fm}}^4$, respectively.

Figure 3

Transition density for the transitions from the ground state in ${}^{16}\mathrm{O}$ calculated with the $\mathrm{VAP}+\mathrm{GCM}$. The scaled transition density $f_{\text{tr}}{\rho }^{(}\mathrm{tr})\left(r\right)$ divided by $\widehat{J_f}\equiv \sqrt{2J_f+1}$ is shown.

Figure 4

Squared charge form factors of ${}^{16}\mathrm{O}$. The theoretical values obtained with the $\mathrm{VAP}+\mathrm{GCM}$ are scaled by $f_{\text{tr}}^2$ consistently with the scaled transition density. For the $1_1^{-}\to 0_1^{+}$ transition, the squared form factors scaled by the modified value $f_{\text{tr}}^2=1.3^2$ of the scaling factor are also shown. The experimental data are from Ref. [48].

Figure 5

$\alpha$-Scattering cross sections on ${}^{16}\mathrm{O}$ at $E_{\alpha }=104$ MeV ($\times {10}^4$), 130 MeV ($\times {10}^2$), 146 MeV, and 386 MeV ($\times {10}^{-2}$). The differential cross sections of the $0_{1,2,3,4,5}^{+}$ obtained by the CC and DWBA calculations are shown by (red) solid and (green) dashed lines, respectively. For the $0_5^{+}$, the cross sections obtained by the CC calculation using the $0_{1,2,3,4,5}^{+}$ and $2_{1,2,...,8}^{+}$ states are also shown by (blue) long-dashed lines. The experimental data at $E_{\alpha }=104$ MeV [43], 130 MeV [12], 146 MeV [44], and 400 MeV [39] are shown by filled circles, open circles, open triangles, and filled squares, respectively. For the $0_3^{+}$ cross sections, the data at $E_{\alpha }=104$ MeV from Ref. [45] and those at $E_{\alpha }=386$ MeV from Ref. [12] are also shown by open squares and open circles, respectively.

Figure 6

$\alpha$-Scattering cross sections on ${}^{16}\mathrm{O}$ at $E_{\alpha }=104$ MeV ($\times {10}^4$), 130 MeV ($\times {10}^2$), 146 MeV, and 386 MeV ($\times {10}^{-2}$). The differential cross sections of the $2_{1,2,3,4}^{+}$ and $4_{1,2}^{+}$ obtained by the CC and DWBA calculations are shown by (red) solid and (green) dashed lines. In the CC calculation for the $4_{1,2}^{+}$ cross sections, the $0_{1,2,3,4}^{+},2_{1,2,...,8}^{+}$, and $4_{1,2,3}^{+}$ states are used. The experimental data at 130 MeV [12], 146 MeV [44], and 386 MeV [12] are shown by open circles, open triangles, and open circles, respectively.

Figure 7

$\alpha$-Scattering cross sections on ${}^{16}\mathrm{O}$ at $E_{\alpha }=104$ MeV ($\times {10}^4$), 130 MeV ($\times {10}^2$), 146 MeV, and 386 MeV ($\times {10}^{-2}$). The differential cross sections of the $3_{1,2}^{-}$ and $1_{1,2}^{-}$ obtained by the default CC and DWBA calculations are shown by (red) solid and (green) dashed lines. The experimental data at 104 MeV [45], 130 MeV [12], 146 MeV [44], and 386 MeV [12] are shown by open squares, open circles, open triangles, and open circles, respectively. For the $1_1^{-}$ states, the result calculated with the modified value $f_{\text{tr}}=1.3$ of the scaling factor for the $0_1^{+}\to 1_1^{-}$ transition density are also shown by (magenta) dotted lines.