Microscopic analysis of extranuclear capture on the<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:msup><mml:mrow/><mml:mrow><mml:mn>16</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mi mathvariant="normal">O</mml:mi><mml:mo>(</mml:mo><mml:mi>p</mml:mi><mml:mo>,</mml:mo><mml:mi>γ</mml:mi><mml:mrow><mml:msup><mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mrow><mml:mn>17</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mi mathvariant="normal">F</mml:mi></mml:math>reaction

Starting from a fully microscopic calculation, the ${}^{16}\mathrm{O}(p,\ensuremath{\gamma}{)}^{17}\mathrm{F}$ radiative-capture reaction is discussed in detail. The generator-coordinate and microscopic $R$-matrix methods are applied to the determination of the bound states and phase shifts of the ${}^{16}\mathrm{O}+p$ system, where ${}^{16}\mathrm{O}$ is described by a closed $p$ shell cluster. The astrophysical $S$ factor is then calculated and compared with experiment. A study of its behavior at very low energies leads to general quantal formulas for the $S$ factor and for its logarithmic derivative at zero energy, which are valid for all cases where capture dominantly occurs when both nuclei are far from each other. The larger capture to the ${1/2}^{+}$ excited state is then explained by its lower binding energy without need for a special halo effect. The logarithmic derivative at zero energy is shown to depend on a slowly varying function of the bound-state Sommerfeld parameter and its different values for capture to the ${5/2}^{+}$ and ${1/2}^{+}$ states are explained. The same expressions are applied to the ${}^{7}\mathrm{Be}(p,\ensuremath{\gamma}{)}^{8}\mathrm{B}$ reaction.

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