Coupled-reaction-channel study of the ${}^{12}\mathrm{C}(\alpha ,{}^8\mathrm{Be})$ reaction and the ${}^8\mathrm{Be}+{}^8\mathrm{Be}$ optical potential

Background: Given the established $2\alpha$ structure of ${}^8\mathrm{Be}$, a realistic model of four interacting $\alpha$ clusters must be used to obtain a ${}^8\mathrm{Be}+{}^8\mathrm{Be}$ interaction potential. Such a four-body problem poses a challenge for the determination of the ${}^8\mathrm{Be}+{}^8\mathrm{Be}$ optical potential (OP) that is still unknown due to the lack of the elastic ${}^8\mathrm{Be}+{}^8\mathrm{Be}$ scattering data.

Purpose: To probe the complex ${}^8\mathrm{Be}+{}^8\mathrm{Be}$ optical potential in the coupled-reaction-channel (CRC) study of the $\alpha$ transfer ${}^{12}\mathrm{C}(\alpha ,{}^8\mathrm{Be})$ reaction measured at $E_{\alpha }=65$ MeV and to obtain the spectroscopic information on the $\alpha +{}^8\mathrm{Be}$ cluster configuration of ${}^{12}\mathrm{C}$.

Method: The three- and four-body continuum-discretized coupled-channel (CDCC) methods are used to calculate the elastic $\alpha +{}^8\mathrm{Be}$ and ${}^8\mathrm{Be}+{}^8\mathrm{Be}$ scattering at the energy around 16 MeV/nucleon, with the breakup effect taken into account explicitly. Based on the elastic cross section predicted by the CDCC calculation, the local equivalent OP's for these systems are deduced for the CRC study of the ${}^{12}\mathrm{C}(\alpha ,{}^8\mathrm{Be})$ reaction.

Results: Using the CDCC-based OP's and $\alpha$ spectroscopic factors given by the cluster model calculation, a good CRC description of the $\alpha$ transfer data for both the ${}^8\mathrm{Be}+{}^8\mathrm{Be}$ and ${}^8\mathrm{Be}+{}^8\mathrm{Be}_{2^{+}}^{*}$ exit channels is obtained without any adjustment of the (complex) potential strength.

Conclusion: The $\alpha +{}^8\mathrm{Be}$ and ${}^8\mathrm{Be}+{}^8\mathrm{Be}$ interaction potential can be described by the three- and four-body CDCC methods, respectively, starting from a realistic $\alpha +\alpha$ interaction. The $\alpha$ transfer ${}^{12}\mathrm{C}(\alpha ,{}^8\mathrm{Be})$ reaction should be further investigated not only to probe of the $4\alpha$ interaction but also the cluster structure of ${}^{12}\mathrm{C}$.

Figure 1

The $\alpha$ configurations and coordinates used for $\alpha +{}^8\mathrm{Be}$ (a) and for ${}^8\mathrm{Be}+{}^8\mathrm{Be}$ (b).

Figure 2

Complex OP for the elastic ${}^8\mathrm{Be}+{}^8\mathrm{Be}$ scattering at $E_{\mathrm{c}.\mathrm{m}.}=41.3$ MeV given by the method (i) and that in the WS form (9) given by the method (ii).

Figure 3

The CDCC prediction for the elastic ${}^8\mathrm{Be}+{}^8\mathrm{Be}$ (upper panel) and $\alpha +{}^8\mathrm{Be}$ (lower panel) scattering at $E_{\mathrm{c}.\mathrm{m}.}=41.3$ and 43.3 MeV, in comparison with the corresponding OM results given by the optical potentials $U_{\mathrm{LEP}}$ and $U_{\mathrm{WSD}}$ determined by the methods (i) and (ii), respectively.

Figure 4

Cluster configurations in the entrance and exit channels of the ${}^{12}\mathrm{C}(\alpha ,{}^8\mathrm{Be}$) reaction, and the corresponding coordinates used for the inputs of the potentials in the CRC calculation.

Figure 5

CRC description of the elastic $\alpha +{}^{12}\mathrm{C}$ scattering at $E_{\alpha }=65$ MeV given by the (unrenormalized) real folded OP and imaginary OP chosen in the WS form [11], in comparison with the data taken from Refs. [23, 24].

Figure 6

CRC description of the $\alpha$ transfer reaction ${}^{12}\mathrm{C}(\alpha ,{}^8\mathrm{Be}){}^8\mathrm{Be}_{\mathrm{g}.\mathrm{s}.}$ measured at $E_{\alpha }=65$ MeV [22], using the $\alpha$ spectroscopic factor $S_{\alpha }\left(\mathrm{g}.\mathrm{s}.\right)\approx 0.36$ taken from the CSM calculation [25]. The CRC results obtained with the CDCC-based optical potentials $U_{\mathrm{WSD}}$ and $U_{\mathrm{LEP}}$ for the ${}^8\mathrm{Be}+{}^8\mathrm{Be}$ partition are shown as the solid and dash-dotted lines, respectively. The dashed line is the CRC result obtained with the WS potential $U_{\mathrm{Int}\mathrm{WS}}$, interpolated from the OP's adopted for the ${}^{7,8}\mathrm{Be}+{}^9\mathrm{Be}$ systems at the nearby energies [5].

Figure 7

The same as Fig. 6 but for the $\alpha$ transfer reaction with one emitting ${}^8\mathrm{Be}$ nucleus being in its $2^{+}$ state ($E_x\approx 2.94$ MeV), using the $\alpha$ spectroscopic factor $S_{\alpha }\left(2^{+}\right)\approx 0.38$ taken from the CSM calculation [25].