Determination of neutron halo radii in the first excited states of ${}^{13}\mathrm{C}$ and ${}^{11}\mathrm{Be}$ with the asymptotic normalization coefficients method

An analysis of the differential cross sections of the ${}^{12}\mathrm{C}(d,p){}^{13}\mathrm{C}$ (ground state, or g.s., $1/2_1^{+}$) and ${}^{10}\mathrm{Be}(d,p){}^{11}\mathrm{Be}$ (g.s., $1/2_1^{-}$) reactions at the incident deuteron energies of 11.8, 25.9, and 30 MeV for the first system, and 12.0 and 25.0 MeV for the second system has been performed by applying the coupled-reaction-channels method. The spectroscopic factors $S_{\text{expt}}$, asymptotic normalization coefficients (ANCs), and rms radii of the last neutron in the ground states and the first excited states of the stable ${}^{13}\mathrm{C}$ and the radioactive ${}^{11}\mathrm{Be}$ nuclei were extracted. The existence of neutron halos in the first excited state of ${}^{13}\mathrm{C}$ and the ground state of ${}^{11}\mathrm{Be}$ is confirmed. The rms radii of halos in these states were found to be $5.72\pm 0.16$ and $8.0\pm 0.2$ fm, respectively, in agreement with experimental results. We showed that a halo is also formed in the 0.32 MeV $1/2_1^{-}$ excited state of ${}^{11}\mathrm{Be}$. The halo radius in this state was determined for the first time and was found to be $R_h=4.65\pm 0.20$ fm.

Figure 1

CRC fit of the elastic cross sections from $d+{}^{12}\text{C}$ scattering at (a) 12.4 MeV, (b) 25.9 MeV, and (c) 29.5 MeV. (a) The solid line represents the CRC calculations, compared with the OM elastic-scattering calculations at 12.4 MeV with the OP from Refs. [53, 55] (dashed curve). The data points are taken from Refs. [37, 53, 54].

Figure 2

CRC fit of the ${}^{12}\mathrm{C}{(d,p)}^{13}\mathrm{C}$ reaction cross sections at (a) $E_{\text{lab}}=11.8$ MeV, (b) $E_{\text{lab}}=25.9$ MeV, and (c) $E_{\text{lab}}=30$ MeV leading to the $1/2^{-}$ ground state. The dashed-dotted, solid, dashed, and dotted lines correspond to the cross sections calculated with different overlap $n+{}^{12}\text{C}$ wave functions generated by the sp potentials with parameter sets numbered 3, 5, 7, and 8 from Table 2. The data points are taken from Refs. [29, 37, 38].

Figure 3

CRC fit of the ${}^{12}\mathrm{C}{(d,p_1)}^{13}\mathrm{C}$ reaction cross sections at (a) $E_{\text{lab}}=11.8$ MeV, (b) $E_{\text{lab}}=30$ MeV and (c) $E_{\text{lab}}=25.9$ MeV leading to the first excited $1/2_1^{+}$ 3.09 MeV state. The dashed, dotted, solid, and dashed-dotted lines correspond to the cross sections calculated with different overlap $n+{}^{12}\text{C}$ wave functions generated by the sp potentials with parameter sets numbered 3, 5, 6, and 7 from Table 3. The data points are taken from Refs. [29, 37, 38].

Figure 4

Examples of the radial sp wave functions $u_{{}^{12}\text{C}n}\left(r\right)$ for the $1/2^{-}\mathrm{g}.\mathrm{s}.$ (top panel) and the first excited 3.09 MeV $1/2_1^{+}$ state (bottom panel) of ${}^{13}\mathrm{C}$ in comparison with the asymptotic parts of the corresponding Hankel functions $h_l\left(i{k}_{{}^{12}\text{C}n}r\right)$ multiplied by $C_{{}^{12}\text{C}n}^{\text{expt}}$.

Figure 5

CRC fit of the elastic cross section from $d+{}^{10}\text{Be}$ scattering at 12.0 MeV (solid line), in comparison with the data from Refs. [39, 68].

Figure 6

CRC fit of the ${}^{10}\text{Be}(d,p){}^{11}\text{Be}$ reaction cross sections at $E_{\text{lab}}=12$ and 25 MeV leading to the $1/2^{+}$ ground state of ${}^{11}\mathrm{Be}$. The lines correspond to the cross sections calculated with different overlap $n+{}^{10}\text{Be}$ wave functions. The dotted and dashed-dotted lines correspond to the parameter sets $r_0=1.25$ fm, $a_0=0.65$ fm, and $r_0=1.6$ fm, $a_0=0.75$ fm, respectively. The solid and dashed lines correspond to the parameter sets numbered 3 and 9 of Table 5. The data points are taken from Refs. [39, 40].

Figure 7

Examples of the radial sp wave functions $u_{{}^{10}\text{Be}n}\left(r\right)$ multiplied by ${\left(S^{\text{expt}}\right)}^{1/2}$ for the $1/2^{+}\mathrm{g}.\mathrm{s}.$ (top panel) and for the first excited $1/2_1^{-}$, 0.32 MeV state (bottom panel) of ${}^{11}\mathrm{Be}$ in comparison with the asymptotic parts of the corresponding Hankel functions $h_l\left(i{k}_{{}^{10}\text{Be}n}r\right)$ multiplied by $C_{{}^{10}\text{Be}n}^{\text{expt}}$.

Figure 8

The same as in Fig. 6, but for the first excited $1/2_1^{-}$, 0.32 MeV state of ${}^{11}\mathrm{Be}$. The dotted, dashed-dotted, solid, and dashed lines correspond to the cross sections calculated with different overlap $n+{}^{10}\text{Be}$ wave functions generated by the sp potentials with parameter sets numbered 1, 2, 4, and 9 of Table 6. The data points are taken from Refs. [39, 40].