Systematic analysis of the peripherality of the ${}^{10}\mathrm{Be}\left(d,p\right){}^{11}\mathrm{Be}$ transfer reaction and extraction of the asymptotic normalization coefficient of ${}^{11}\mathrm{Be}$ bound states

We reanalyze the experiment of Schmitt et al. on the ${}^{10}\mathrm{Be}(d,p){}^{11}\mathrm{Be}$ transfer reaction [Phys. Rev. Lett. 108, 192701 (2012)] by exploring the beam-energy and angular ranges at which the reaction is strictly peripheral. We consider the adiabatic distorted wave approximation (ADWA) to model the reaction and use a Halo-EFT description of ${}^{11}\mathrm{Be}$ to systematically explore the sensitivity of our calculations to the short-range physics of the ${}^{10}\mathrm{Be}-n$ wave function. We find that by selecting the data at low beam energy and forward scattering angle the calculated cross sections scale nearly perfectly with the asymptotic normalization coefficient (ANC) of the ${}^{11}\mathrm{Be}$ bound states. Following these results, a comparison of our calculations with the experimental data gives a value of $C_{1s1/2}=0.785\pm 0.03{\mathrm{fm}}^{-1/2}$ for the ${\frac{1}{2}}^{+}$ ground-state ANC and $C_{0p1/2}=0.135\pm 0.005{\mathrm{fm}}^{-1/2}$ for the ${\frac{1}{2}}^{-}$ excited state, which are in perfect agreement with the ab initio calculations of Calci et al., who obtain $C_{1/2^{+}}^{\mathit{ab}\mathit{initio}}=0.786{\mathrm{fm}}^{-1/2}$ and $C_{1/2^{-}}^{\mathit{ab}\mathit{initio}}=0.129{\mathrm{fm}}^{-1/2}$ [Phys. Rev. Lett. 117, 242501 (2016)].

Figure 1

Illustration of the three-body system with associated coordinates.

Figure 2

Reduced radial wave functions $u_{1s1/2}$ of the ${\frac{1}{2}}^{+}$ g.s. of ${}^{11}\mathrm{Be}$ obtained with the nine Gaussian potentials of Table 1.

Figure 3

Reduced radial wave functions $u_{0p1/2}$ of the ${\frac{1}{2}}^{-}$ e.s. of ${}^{11}\mathrm{Be}$ obtained with the nine Gaussian potentials of Table 1.

Figure 4

Analysis of the differential cross sections of ${}^{10}\mathrm{Be}(d,p){}^{11}\mathrm{Be}$(g.s.) for a deuteron energy $E_d=21.4\mathrm{MeV}$. The results of the ADWA calculations are presented for every potential of Table 1.

Figure 5

Same as Fig. 4 for $E_d=18\mathrm{MeV}$.

Figure 6

Same as Fig. 4 for $E_d=15\mathrm{MeV}$.

Figure 7

Same as Fig. 4 for $E_d=12\mathrm{MeV}$.

Figure 8

ANCs extracted for the ground state of ${}^{11}\mathrm{Be}$ by minimizing the ${\chi }^2$ (7) for each beam energy and each potential of Table 1. The ab initio result ($C_{1/2^{+}}^{\mathit{ab}\mathit{initio}}=0.786{\mathrm{fm}}^{-1/2}$) is displayed for comparison by the dashed line.

Figure 9

The angular distribution for ${}^{10}\mathrm{Be}(d,p){}^{11}\mathrm{Be}$(g.s.) at all experimental energies after scaling to the ANC obtained by the ${\chi }^2$ minimization $C_{1s1/2}=0.785{\mathrm{fm}}^{-1/2}$.

Figure 10

ANCs extracted for the ${}^{11}\mathrm{Be}$ e.s. by minimizing the ${\chi }^2$ (7) for each beam energy and each potential of Table 1. The ab initio result ($C_{1/2^{-}}^{\mathit{ab}\mathit{initio}}=0.129{\mathrm{fm}}^{-1/2}$) is displayed for comparison.

Figure 11

Influence of the nucleon-nucleus optical potential on the transfer cross section for ${}^{10}\mathrm{Be}(d,p){}^{11}\mathrm{Be}$(g.s.) at $E_d=12\mathrm{MeV}$. The Gaussian ${}^{10}\mathrm{Be}-n$ potential is chosen with a width $r_0=1.4\mathrm{fm}$ in both cases.