Four-body extension of the continuum-discretized coupled-channels method

I develop an extension of the continuum-discretized coupled-channels (CDCC) method to reactions where both nuclei present a low breakup threshold. This leads to a four-body model, where the only inputs are the interactions describing the colliding nuclei, and the four optical potentials between the fragments. Once these potentials are chosen, the model does not contain any additional parameter. First I briefly discuss the general formalism, and emphasize the need for dealing with large coupled-channel systems. The method is tested with existing benchmarks on $4\alpha$ bound states with the Ali-Bodmer potential. Then I apply the four-body CDCC to the ${}^{11}\mathrm{Be}+d$ system, where I consider the ${}^{10}\mathrm{Be}(0^{+},2^{+})+n$ configuration for ${}^{11}\mathrm{Be}$. I show that breakup channels are crucial to reproduce the elastic cross section, but that core excitation plays a weak role. The ${}^7\mathrm{Li}+d$ system is investigated with an $\alpha +t$ cluster model for ${}^7\mathrm{Li}$. I show that breakup channels significantly improve the agreement with the experimental cross section, but an additional imaginary term, simulating missing transfer channels, is necessary. The full CDCC results can be interpreted by equivalent potentials. For both systems, the real part is weakly affected by breakup channels, but the imaginary part is strongly modified. I suggest that the present wave functions could be used in future DWBA calculations.

Figure 1

Cluster configuration and coordinates used in the four-body model.

Figure 2

Energy of the $4\alpha$ system with the Ali-Bodmer potential [27] for $J=0^{+}$ (upper panel) and $J=2^{+}$ (lower panel), by neglecting the Coulomb potential. The labels indicate ${\ell }_{\max }$, the maximum angular momentum in the $\alpha +\alpha$ system. The ${}^8\mathrm{Be}+{}^8\mathrm{Be}$ and $\alpha +{}^{12}\mathrm{C}$ thresholds are shown on the right of the figure.

Figure 3

See caption of Fig. 2 for $J=0^{+}$ and by including the Coulomb $\alpha +\alpha$ potential (22).

Figure 4

${}^{11}\mathrm{Be}+p$ elastic scattering (a) and breakup (b) cross sections at a ${}^{11}\mathrm{Be}$ energy $E=26.9A\mathrm{MeV}$ (the elastic cross section is divided by the Rutherford cross section). The solid and dashed lines are obtained with and without core excitation, respectively. The experimental data are taken from Ref. [35].

Figure 5

${}^{11}\mathrm{Be}+d$ elastic cross section (divided by the Rutherford cross section) at $E\left({}^{11}\mathrm{Be}\right)=26.9A\mathrm{MeV}$, with the single-channel approximation, and with all breakup channels. The solid lines are obtained with core excitation, and the dashed line with the ${}^{10}\mathrm{Be}\left(0^{+}\right)+n$ configuration only. Tha data are taken from Ref. [11].

Figure 6

Convergence of the ${}^{11}\mathrm{Be}+d$ elastic cross section as a function of the truncation angular momentum ${\ell }_{\max }$ in the deuteron (a) and in ${}^{11}\mathrm{Be}$ (c), and of the maximum pseudostate energy $E_{\max }$ (b), (d). In some cases, the curves are superimposed, and a double label is used.

Figure 7

Amplitude of the scattering matrix $|U_{11}^{J\pi }|$ for $L=J-1/2$, and for different conditions of calculation (see text). Labels “gs” and “BU” refer to the ground state and to breakup channels, respectively.

Figure 8

${}^{11}\mathrm{Be}+d$ breakup cross sections with core excitation (solid lines) and without core excitation (dashed lines). The data are taken from Ref. [11].

Figure 9

Ratio to the Rutherford cross section for $t+p$ elastic scattering at $E_p=12\mathrm{MeV}$ (see text). The data are taken from Ref. [42].

Figure 10

Ratio to the Rutherford cross section for ${}^7\mathrm{Li}+d$ elastic scattering at $E_d=25\mathrm{MeV}$ with different conditions of calculation (see text). The dashed lines are obtained by adding the imaginary potential (25). The data are taken from Ref. [37].

Figure 11

(a) ${}^{11}\mathrm{Be}+d$ equivalent potentials (30). (b) Elastic cross sections (divided by the Rutherford cross sections) with potential (30) (solid lines) and with the CDCC calculation (dashed lines). The single-channel curves are in red, and the full calculations in black.

Figure 12

See caption of Fig. 11 for the ${}^7\mathrm{Li}+d$ system. Results with the imaginary potential (25) (using $N_I=0.2$) are also shown. Notice that the imaginary potential for the single-channel calculation is exactly zero.