Breakup and finite-range effects on the ${}^{8}B{(d,n)}^9\text{C}$ reaction

The astrophysical factor of ${}^8\mathrm{B}{(p,\gamma )}^9\mathrm{C}$ at zero energy, $S_{18}\left(0\right)$, is determined by a three-body coupled-channels analysis of the transfer reaction ${}^8\mathrm{B}{(d,n)}^9\mathrm{C}$ at 14.4 MeV/nucleon. Effects of the breakup channels of $d$ and ${}^9\mathrm{C}$ are investigated with the continuum-discretized coupled-channels method. It is found that, in the initial and final channels, respectively, the transfer process through the breakup states of $d$ and ${}^9\mathrm{C}$, its interference with that through their ground states in particular, gives a large increase in the transfer cross section. The finite-range effects with respect to the proton-neutron relative coordinate are found to be about 20%. As a result of the present analysis, $S_{18}\left(0\right)=22\pm 6\mathrm{eV}\mathrm{b}$ is obtained, which is smaller than the result of the previous distorted-wave Born approximation analysis by about 51%.

Figure 1

Illustration of the three-body system.

Figure 2

Cross section of the transfer reaction ${}^8\mathrm{B}{(d,n)}^9\mathrm{C}$ at 14.4 MeV/nucleon as a function of the neutron emission angle in the c.m. frame. The result of the CCBA (solid line) is normalized to the experimental data [4].

Figure 3

Breakup effects of $d$ and ${}^9\mathrm{C}$ on the cross section of ${}^8\mathrm{B}{(d,n)}^9\mathrm{C}$ at 14.4 MeV/nucleon. The thick solid and thin solid lines show, respectively, the results with and without the breakup states of both $d$ and ${}^9\mathrm{C}$. The dashed (dotted) line represents the result with the breakup states of $d$ (${}^9\mathrm{C}$) in the transition matrix $T_{\beta \alpha }$ being neglected. The cross section corresponding to the ET is shown by the dash-dotted line. See the text for detail.

Figure 4

(a) $D_{pn}^i$ for several $i\text{th}$ states with the eigenenergy ${\varepsilon }_{pn}^i$. (b) Same as in panel (a) but for the $\alpha -d$ system.

Figure 5

CCBA results of the FR calculation (solid line), the ZR calculation (dotted line), and the ZR calculation with the FRC (dashed line).

Figure 6

(a) Real and (b) imaginary parts of the correction function $F_{\mathrm{LEA}}^i$ defined by Eq. (33). Each line corresponds to the result with ${\varepsilon }_{pn}^i$ specified in the legends.

Figure 7

$S_{18}\left(0\right)$ in the present work (circle) compared with the results evaluated from the ${}^8\mathrm{B}{(d,n)}^9\mathrm{C}$ reaction (diamond) [4] and values extracted from ${}^9\mathrm{C}$ breakup reactions (triangle [2], cross [3], and square [5]).