Observation of a breakup-induced $\alpha$-transfer process for some bound states of ${}^{16}\mathrm{O}$ populated by the ${}^{12}\mathrm{C}$(${}^6\mathrm{Li}$,$d$)${}^{16}\mathrm{O}$${}^{*}$ reaction

Background: The ${}^{12}\mathrm{C}{(}^6\mathrm{Li},d)$ reaction has been used as an indirect method to calculate the astrophysical $S$ factor for the ${}^{12}\mathrm{C}$($\alpha ,\gamma )$ reaction at Gamow energy (300 keV).

Purpose: The ${}^{12}\mathrm{C}{(}^6\mathrm{Li},d)$ reaction is usually interpreted in terms of direct transfer. In this work we investigate the reaction mechanism and determine the effects of breakup on transfer and therefore on the extracted spectroscopic amplitudes.

Method: The deuteron angular distributions for the ${}^{12}\mathrm{C}{{(}^6\mathrm{Li},d)}^{16}{\mathrm{O}}^{*}$has been measured at 20 MeV, populating discrete states of ${}^{16}\mathrm{O}$. continuum discretized coupled channel-coupled reaction channel (CDCC-CRC) calculations have been used to analyze the data.

Results: Results show a new reaction mechanism, where transfer occurs after the breakup of the loosely bound ${}^6\mathrm{Li}$ in the population of some bound states of ${}^{16}\mathrm{O}$. A comparison of the CDCC-CRC calculations with respect to the measured data were used to determine the $\alpha$ spectroscopic amplitudes and factors for the different states of ${}^{16}\mathrm{O}$. Using the spectroscopic amplitudes obtained in this work, the $E2$ $S$ factor for the ${}^{12}\mathrm{C}$($\alpha$,$\gamma$) reaction has been calculated in the framework of a two-body potential model and compared to measurements.

Conclusions: The present study very clearly shows that the breakup and transfer coupling effects are strong in the ${}^{12}\mathrm{C}{(}^6\mathrm{Li},d)$ reaction. The present work extracts, in the framework of a coupled reaction channel theory, the spectroscopic amplitudes of the bound and unbound states of ${}^{16}\mathrm{O}$. All previous analysis and new measurements should therefore be reexamined from this viewpoint to extract the astrophysical observables correctly.

Figure 1

The measured deuteron spectra at 45${}^{\circ }$ showing both the discrete peaks and the continuous bump. A double Gaussian fit to obtain the separate contribution of the 6.92- and 7.12-MeV states is shown in the inset.

Figure 2

The comparison of the measured angular distributions for the $\alpha$-bound states of ${}^{16}\mathrm{O}$ with $R$-matrix FRDWBA calculations (green dashed lines), Hauser-Feshbach calculations (black dotted lines), and CDCC-CRC (red solid lines) calculations, where all the bound states and unbound resonance states of ${}^{16}\mathrm{O}$ are considered. The blue solid lines denote the CDCC-CRC calculations summed to the CN component. The black solid lines are the CRC calculations without the breakup channels. The open box symbols are the angular distribution data of Ref. [10].

Figure 3

The comparison of the measured angular distributions for the $\alpha$ unbound states of ${}^{16}\mathrm{O}$ with $R$-matrix FRDWBA calculations (green dashed lines), Hauser-Feshbach calculations (black dotted lines), and CDCC-CRC (red solid lines) calculations, where all the bound states and unbound resonance states of ${}^{16}\mathrm{O}$ are considered. The blue solid lines denote the CDCC-CRC calculations summed to the CN component. The 8.87(2${}^{-}$)-MeV unnatural parity state is interpreted only in terms of Hauser-Feshbach calculation as this state cannot be populated from direct reactions. The black lines are the CRC calculations without the breakup states. The open box symbols are the same as in Ref. [10].

Figure 4

The phase shift as a function of the excitation energies of (a) ${}^{16}\mathrm{O}$ generated by the Gaussian $\alpha +{}^{12}\mathrm{C}$ binding potential [18] and (b) ${}^6\mathrm{Li}$ generated by the $\alpha +d$ potential [19]. The three resonances of ${}^{16}\mathrm{O}$ and ${}^6\mathrm{Li}$ are shown by solid, dashed, and dotted lines. For details, see text.

Figure 5

The ratio of the ${}^6\mathrm{Li}+{}^{12}\mathrm{C}$ elastic scattering cross-section data to the Rutherford cross section plotted vs scattering angle. The experimental data are taken from Ref. [24] measured by Ref. [20]. The red line shows the calculation using CDCC-CRC theory with the imaginary part of the $\alpha +{}^{12}\mathrm{C}$ and $d+{}^{12}\mathrm{C}$ potential parameters as given in Table 1 renormalized to fit the data. The black line shows the CRC calculation without the breakup channels of ${}^6\mathrm{Li}$ and requires a different renormalization for the imaginary depth of the $\alpha +{}^{12}\mathrm{C}$ and $d+{}^{12}\mathrm{C}$ potentials to fit the experimental data.

Figure 6

Comparison of potential model calculations of $E2$ $S$ factor with measured data [30, 31]. The solid red line and the blue long-dashed line represent the calculations using the potentials which reproduce the features of the 6.92-MeV and the 9.85-MeV 2${}^{+}$ states, respectively. The spectroscopic amplitudes of the g.s. and 6.92-MeV states that go into this calculation are those evaluated in this work (Table 2). The black dashed lines show the value for the $S$ factor owing to the lower and upper limits of ground-state $S_{\alpha }$ from Table 2 (for details, see text).