Potential model for nuclear astrophysical fusion reactions with a square-well potential

The potential model for nuclear astrophysical reactions requires a considerably shallow nuclear potential when a square-well potential is employed to fit experimental data. We discuss the origin of this apparently different behavior from that obtained with a smooth Woods-Saxon potential, for which a deep potential is often employed. We argue that due to the sharp change of the potential at the boundary the radius parameter tends to be large in the square-well model, which results in a large absorption radius. The wave function then needs to be suppressed in the absorption region, which can eventually be achieved by using a shallow potential. We thus clarify the reason why the square-well potential has been able to reproduce a large amount of fusion data.

Figure 1

(a) The astrophysical $S$ factor for the ${}^{16}\mathrm{O}+{}^{16}\mathrm{O}$ reaction. The solid and the dashed lines are obtained with a square-well and a Woods-Saxon potential, respectively. The experimental data are taken from Ref. [14]. (b) The radial dependence of the total potentials used in the calculations shown in (a).

Figure 2

Same as Fig. 1, but for two different depths of the Woods-Saxon potential as indicated in the figure.

Figure 3

(a) Comparison of the astrophysical $S$ factor for the ${}^{16}\mathrm{O}+{}^{16}\mathrm{O}$ system obtained with two different depth parameters of the square-well potential. The solid line is the same as that in Fig. 1 and is obtained with $V_0=9.4\mathrm{MeV}$, while the dot-dashed line is obtained with $V_0=0\mathrm{MeV}$. The range parameter of the square-well potential is set to be $R=8.13\mathrm{fm}$ for both cases. The experimental data are taken from Ref. [14]. (b) The radial dependence of the square-well potentials used to compute the $S$ factors shown in (a). The radial wave functions (in arbitrary units) at $E=7\mathrm{MeV}$ for $l=0$ are also shown by the thin solid (for $V_0=9.4\mathrm{MeV}$) and the thin dashed (for $V_0=0\mathrm{MeV}$) lines.

Figure 4

Same as Fig. 3, but for the case where the range parameter of the imaginary part of the square-well potential is set independently to that for the real part. Those are taken to be 8.13 and 6.5 fm for the real and the imaginary parts, respectively. The solid line is obtained with $V_0=9.4\mathrm{MeV}$, while the dotted line is obtained with $V_0=7.0\mathrm{MeV}$. The experimental data are taken from Ref. [14].