Microscopic calculation of inelastic proton scattering off ${}^{18}\mathrm{O}, {}^{10}\mathrm{Be}, {}^{12}\mathrm{Be},$ and ${}^{16}\mathrm{C}$ to study neutron excitation in neutron-rich nuclei

Microscopic coupled-channel calculations of inelastic proton scattering are performed to study neutron excitations in the $2_1^{+}$ states of ${}^{18}\mathrm{O}$, ${}^{10}\mathrm{Be}$, ${}^{12}\mathrm{Be}$, and ${}^{16}\mathrm{C}$. The proton-nucleus potentials in the coupled-channel calculation are microscopically derived by folding the Melbourne $g$-matrix $NN$ interaction with the matter and transition densities of the target nuclei obtained via structure model calculations of the antisymmetrized molecular dynamics. The calculated results reasonably reproduce the elastic and inelastic proton scattering cross section, and support the dominant contribution of neutrons in the $2_1^{+}$ excitation of ${}^{12}\mathrm{Be}$ and ${}^{16}\mathrm{C}$ as well as ${}^{18}\mathrm{O}$. The sensitivity of the inelastic scattering cross sections to the neutron transition density is discussed as well as the exotic behavior of the neutron transition density with amplitude in the outer regions of ${}^{12}\mathrm{Be}$ and ${}^{16}\mathrm{C}$.

Figure 1

Cross sections of the elastic and inelastic proton scattering off ${}^{12}\mathrm{C}$ at $E_p=35\mathrm{MeV}$, $E_p=65\mathrm{MeV}(\times {10}^{-2})$, and $E_p=135\mathrm{MeV}(\times {10}^{-4})$ calculated using the $\mathrm{AMG}+\mathrm{GCM}$ and RGM densities. The results of the CC and DWBA calculations with the $\mathrm{AMD}+\mathrm{GCM}$ densities and the CC calculation with the RGM densities are shown by the red solid, blue dotted, and magenta dashed lines, respectively. The experimental data are from Refs. [71, 72, 73, 74, 75].

Figure 2

Cross sections of the elastic and inelastic proton scattering off ${}^{16}\mathrm{O}$ at $E_p=35\mathrm{MeV}$ and $E_p=135\mathrm{MeV}(\times {10}^{-2})$ calculated using the $\mathrm{AMG}+\mathrm{GCM}$ densities. The results of the CC and DWBA calculations are shown by the red solid and blue dotted lines, respectively. The experimental data are from Refs. [71, 76, 77].

Figure 3

Neutron and proton densities of ${}^{18}\mathrm{O}$: (a) the neutron and proton matter densities of the ground state, (b) the neutron and proton transition densities for the $0_1^{+}\to 2_1^{+}$ transition, and (c) the $r^2$-weighted neutron transition density calculated with AMD. The renormalized proton and neutron transition densities adjusted to the experimental $B\left(E2\right)$ of ${}^{18}\mathrm{O}$ and that of ${}^{18}\mathrm{Ne}$ are shown, respectively. The experimental neutron transition density ${\rho }_{n,\exp }^{\mathrm{tr}(p,p^{\prime })}$ reduced from the $(p,p^{\prime })$ scattering at $E=135\mathrm{MeV}/\mathrm{u}$ [19] and the experimental proton transition density ${\rho }_{p,\exp }^{\mathrm{tr}(e,e^{\prime })}$ measured with the electron scattering data [83] are also shown.

Figure 4

Elastic and inelastic form factors of ${}^{18}\mathrm{O}$. The inelastic form factors for $0_1^{+}\to 2^{+}$ are the renormalized ones adjusted to the experimental $B\left(E\lambda \right)$. The experimental data measured by the electron scattering are from Ref. [83].

Figure 5

Cross sections of the elastic and inelastic proton scattering off ${}^{18}\mathrm{O}$ at $E=24\mathrm{MeV}/\mathrm{u}$ ($\times 10$), 35 MeV/u, 43 MeV/u ($\times {10}^{-1}$), and 135 MeV/u $(\times {10}^{-2})$ calculated using the default AMD densities (solid lines). The experimental data for $E=43\mathrm{MeV}/\mathrm{u}$ are the cross sections measured in inverse kinematics. For the $2_1^{+}$ cross sections, the calculated result with the experimental neutron transition density ${\rho }_{n,\exp }^{\mathrm{tr}(p,p^{\prime })}\left(r\right)$ and that with the reduced neutron transition density $0.88{\rho }_n^{\mathrm{tr}}\left(r\right)$ are also shown by the dashed and dotted lines, respectively. The experimental data are from Refs. [19, 21, 71, 84].

Figure 6

Cross sections of (a) elastic and (b) inelastic neutron scattering off ${}^{18}\mathrm{O}$ at $E=24\mathrm{MeV}/\mathrm{u}$. For the $2_1^{+}$ cross sections, the CC calculation with the experimental neutron transition density ${\rho }_{n,\exp }^{\mathrm{tr}(p,p^{\prime })}\left(r\right)$ and that with $0.88{\rho }_n^{\mathrm{tr}}\left(r\right)$ are also shown by the blue dotted and light-blue dashed lines, respectively, in addition to that with the default AMD densities (the red solid lines). The data are from Ref. [18].

Figure 7

Comparison of the CC and DWBA calculations of the proton scattering off ${}^{18}\mathrm{O}$ with the default ${\rho }_n^{\mathrm{tr}}\left(r\right)$. The (a) elastic and (b) inelastic cross sections at $E=24.5\mathrm{MeV}/\mathrm{u}$ and 35 MeV/u are shown in comparison to the experimental data [71, 84].

Figure 8

Neutron and proton matter densities of the ground states of (a) ${}^{10}\mathrm{Be}$, (b) ${}^{12}\mathrm{Be}$, and (c) ${}^{16}\mathrm{C}$ calculated with AMD.

Figure 9

Neutron (${\rho }_n^{\mathrm{tr}}$) and proton (${\rho }_p^{\mathrm{tr}}$) transition densities for $0_1^{+}\to 2_1^{+}$ for (a) ${}^{10}\mathrm{Be}$, (b) ${}^{12}\mathrm{Be}$, and (c) ${}^{16}\mathrm{C}$ calculated with AMD. The proton and neutron transition densities of ${}^{10}\mathrm{Be}$ are renormalized to adjust the experimental $B\left(E2\right)$ values of ${}^{10}\mathrm{Be}$ and ${}^{10}\mathrm{C}$, respectively.

Figure 10

Cross sections of elastic proton scattering off (a) ${}^{10}\mathrm{Be}$ at $E=60\mathrm{MeV}/\mathrm{u}$, (b) ${}^{12}\mathrm{Be}$ at $E=55\mathrm{MeV}/\mathrm{u}$, and (c) ${}^{10}\mathrm{C}$ at $E=45\mathrm{MeV}/\mathrm{u}$ according to CC calculations with the AMD densities (red solid lines). The one-step cross sections obtained by the DWBA calculation are also shown (blue dotted lines). The calculations are compared to the experimental data measured in inverse kinematics of ${}^{10}\mathrm{Be}$ at 59.2 MeV/u [85], ${}^{12}\mathrm{Be}$ at 55 MeV/u [86], and ${}^{10}\mathrm{C}$ at 45.3 MeV/u [7].

Figure 11

Cross sections of inelastic proton scattering to the $2_1^{+}$ state of (a) ${}^{10}\mathrm{Be}$ at $E=60\mathrm{MeV}/\mathrm{u}$, (b) ${}^{12}\mathrm{Be}$ at $E=55\mathrm{MeV}/\mathrm{u}$, and (c) ${}^{10}\mathrm{C}$ at $E=45\mathrm{MeV}/\mathrm{u}$ according to CC calculations (red solid lines). The one-step cross sections obtained by the DWBA calculation are also shown (blue dotted lines). In panel (b) for ${}^{12}\mathrm{Be}$, the DWBA calculation using the neutron transition density ${\rho }_n^{\mathrm{tr}}\left(r\right)=(M_n/M_p){\rho }_p^{\mathrm{tr}}\left(r\right)$ is also shown for comparison (the magenta dash-dotted line). The calculations are compared to the experimental data measured in inverse kinematics of ${}^{10}\mathrm{Be}$ at 59.2 MeV/u [3], ${}^{12}\mathrm{Be}$ at 53.8 MeV/u [3], and ${}^{10}\mathrm{C}$ at 45.3 MeV/u [7]. For the inelastic scattering of ${}^{10}\mathrm{Be}({}^{12}\mathrm{Be}$), ${\theta }_{\text{lab}}$ is kinematically limited within 5.6 (4.7) degrees; however, the data contain effects due to the finite size and angular spread of the incident beam, multiple scatterings in the target, and the detector geometry.

Figure 12

(a) Cross sections of elastic proton scattering off ${}^{16}\mathrm{C}$ at $E=50\mathrm{MeV}/\mathrm{u}$ and (b) those of inelastic scattering at $E=33\mathrm{MeV}/\mathrm{u}$ according to CC calculations with the AMD densities (red solid lines). The one-step cross sections obtained via the DWBA calculation are also shown (blue dotted lines). The calculations are compared to the experimental data [9, 87] measured in inverse kinematics. For the inelastic scattering, ${\theta }_{\text{lab}}$ is kinematically limited within 3.6 degrees; however, the data contain effects due to the finite size and angular spread of the incident beam, multiple scatterings in the target, and the detector geometry.

Figure 13

$b_n^{(p,p^{\prime })}/b_p^{(p,p^{\prime })}$ ratio of the proton scattering at $E=30$, 45, 60, 80, 100, and 120 MeV/u. The values for ${}^{18}\mathrm{O}$ (magenta open squares), ${}^{10}\mathrm{Be}$ (green crosses), ${}^{12}\mathrm{Be}$ (red filled circles), and ${}^{16}\mathrm{C}$ (blue filled triangles) were calculated with the default densities, and the values for ${}^{12}\mathrm{Be}$ (red open circles), and ${}^{16}\mathrm{C}$ (blue open triangles) for the ${\rho }_n^{\mathrm{tr}}\left(r\right)=(M_n/M_p){\rho }_p^{\mathrm{tr}}\left(r\right)$ case are shown.