Zero-energy determination of the astrophysical<b><i>S</i></b>factor and effective-range expansions

The first three terms of a Taylor expansion of the astrophysical S factor are determined in the potential model of radiative-capture reactions. As input, the radial Schr\"odinger equation and its inhomogeneous energy derivatives are solved at zero energy. The radial wave function and its energy derivatives are obtained by matching the solutions of these equations with the corresponding exact asymptotic forms. Explicit expressions are derived for the S factor and for its first and second derivatives at zero energy. The same algorithm allows one to accurately determine the first terms of the effective-range expansion. In particular, the effective-range formula converges much faster than the Schwinger-Bethe formula. The method is illustrated with potential-model descriptions of the ${}^{3}\mathrm{He}(\ensuremath{\alpha},\ensuremath{\gamma}{)}^{7}\mathrm{Be},$ ${}^{6}\mathrm{Li}(p,\ensuremath{\gamma}{)}^{7}\mathrm{Be},$ ${}^{7}\mathrm{Be}(p,\ensuremath{\gamma}{)}^{8}\mathrm{B},$ and ${}^{16}\mathrm{O}(p,\ensuremath{\gamma}{)}^{17}\mathrm{F}$ reactions.

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