Experimental study of the low-lying negative-parity states in ${}^{11}\mathrm{Be}$ using the ${}^{12}\mathrm{B}(d,{}^3\mathrm{He}){}^{11}\mathrm{Be}$ reaction

Low-lying negative-parity states in ${}^{11}\mathrm{Be}$ having dominant $p$-wave neutron configurations were studied using the ${}^{12}\mathrm{B}(d,{}^3\mathrm{He}){}^{11}\mathrm{Be}$ proton-removal reaction in inverse kinematics. The $1/2_1^{-}$ state at 0.32 MeV, the $3/2_1^{-}$ state at 2.56 MeV, and one or both of the states including the $5/2_1^{-}$ level at 3.89 MeV and the $3/2_2^{-}$ level at 3.96 MeV were populated in the present reaction. Spectroscopic factors were determined from the differential cross sections using a distorted wave Born approximation method. The $p$-wave proton removal strengths were well described by the shell model calculations while the Nilsson model calculation underestimates the spectroscopic factors for the higher excited states. Results from both variational Monte Carlo and no-core shell-model calculations were also compared with the experimental observations.

Figure 1

The $\mathrm{\Delta }E-E$ spectrum obtained using one of the recoil detector telescopes with ${}^{12}\mathrm{B}$ incident on the ${\left({\mathrm{CD}}_2\right)}_n$ target. The data shown required a coincidence with a particle in the PSD array. The particle groups labeled ${}^{11}\mathrm{Be}\left({}^{10}\mathrm{Be}\right)$ and ${}^{12}\mathrm{B}$ are from neutron bound (unbound) states in ${}^{11}\mathrm{Be}$ and the elastic scattering of ${}^{12}\mathrm{B}$, respectively.

Figure 2

Measured ${}^3\mathrm{He}$ energies ($E$) as a function of the distance from the target ($Z$) for the ${}^{12}\mathrm{B}(d,{}^3\mathrm{He}){}^{11}\mathrm{Be}$ reaction in inverse kinematics at 12 MeV/u with a magnetic-field strength of 2.3 T. The data shown required a coincidence with either (a) ${}^{11}\mathrm{Be}$ or (b) ${}^{10}\mathrm{Be}$ recoils as shown in Fig. 1. Final states identified in ${}^{11}\mathrm{Be}$ are labeled by their corresponding excitation energies. (c) The simulation for the different excited states in the ${}^{12}\mathrm{B}\left(d{,}^3\mathrm{He}\right)$ reaction. See details in the text.

Figure 3

The excitation-energy spectrum of ${}^{11}\mathrm{Be}$ neutron bound (blue solid line) and unbound (red dotted line) states determined from the dataset presented in Figs. 2 and 2, respectively. States identified in the present paper are labeled with their corresponding excitation energies.

Figure 4

Experimental (black points) and calculated (red solid lines) angular distributions for the (a) 0.32-, (b) 2.65-, and (c) 3.89-MeV transitions in the ${}^{12}\mathrm{B}(d,{}^3\mathrm{He}){}^{11}\mathrm{Be}$ reaction. The curves represent distorted-wave Born 260 approximation (DWBA) calculations for $\ell =1$ transfer. Only statistical uncertainties are shown for the experimental data, and there is a systematic uncertainty of $\approx 30\%$ on the absolute cross-section scale. The geometrical acceptance of the ${}^{10}\mathrm{Be}$ recoils for the neutron-unbound states of ${}^{11}\mathrm{Be}$ is plotted as black dashed curves.

Figure 5

The (a) experimental and (b)–(d) calculated excitation energies and spectroscopic factors of the $1/2_1^{-},3/2_1^{-}$, and $5/2_1^{-}$ states of ${}^{11}\mathrm{Be}$ from the ${}^{12}\mathrm{B}(d,{}^3\mathrm{He}){}^{11}\mathrm{Be}$ reaction (slash bars) and $0_1^{+}$ and $2_1^{+}$ states of ${}^{10}\mathrm{Be}$ from the ${}^{11}\mathrm{B}(d,{}^3\mathrm{He}){}^{10}\mathrm{Be}$ reaction (dotted bars). Results shown in panels (b)–(d) were calculated using the shell model with the YSOX interaction [34], the VMC method [23], and the Nilsson model [35], respectively. The error bars for the experimental values are just for relative $S$. The blue dashed line in (a) is the $(2j+1)$-weighted energy centroid of $3/2_1^{-}$ and $5/2_1^{-}$ states in ${}^{11}\mathrm{Be}$. Note that the spectroscopic factors and excitation energies of the first excited state in (a)–(d) were normalized to unity and the experimental value ($E_x=0.32\mathrm{MeV}$), respectively.

Figure 6

Ab initio NCCI calculated energy spectrum for negative-parity states of ${}^{11}\mathrm{Be}$ with the Daejeon16 interaction. Energies are plotted against an angular momentum axis scaled as $J(J+1)$ as appropriate for rotational analysis. (a) Calculated negative-parity spectrum ($N_{\max }=10,\hslash \omega =15\mathrm{MeV}$), shown with fits of the rotational energy formula (1) to the calculated band member energies (lines). States are classified as $0\hslash \omega$ (shaded square) or “$2\hslash \omega$” (open squares) as described in the text. (b) Calculated relative energies, taken with respect to the $1/2_1^{-}$ ground state of the negative-parity space. These are shown for successively larger bases as indicated by increasing symbol size from $N_{\max }=4$ (dotted line) through 10 (solid line). The relative energy of the calculated $1/2_1^{+}$ is also shown (diamonds) from $N_{\max }=5$ through 11. Energies for the experimental counterparts are shown ($-$ for negative parity or $+$ for positive parity) for comparison (these are labeled with the experimental excitation energies in MeV for convenient identification).

Figure 7

Decomposition of NCCI calculated eigenstates for the (a) $5/2_2^{-}$ and (b) $5/2_1^{-}$ states with respect to the number of excitation quanta $N_{\mathrm{ex}}$ in the contributing oscillator configurations. These decompositions are for the same calculations as shown in Fig. 6 with the histograms overlaid for $N_{\max }=4$ (dotted line) through 10 (solid line).