Asymptotic normalization coefficients for<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:msup><mml:mrow/><mml:mrow><mml:mn>8</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:msup><mml:mover><mml:mrow><mml:mi mathvariant="normal">B</mml:mi></mml:mrow><mml:mrow><mml:mo>→</mml:mo></mml:mrow></mml:mover><mml:mrow><mml:mn>7</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mi mathvariant="normal">Be</mml:mi><mml:mo>+</mml:mo><mml:mi>p</mml:mi></mml:math>from a study of<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:msup><mml:mrow/><mml:mrow><mml:mn>8</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:msup><mml:mover><mml:mrow><mml:mi mathvariant="normal">Li</mml:mi></mml:mrow><mml:mrow><mml:mo>→</mml:mo></mml:mrow></mml:mover><mml:mrow><mml:mn>7</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mi mathvariant="normal">Li</mml:mi><mml:mo>+</mml:mo><mml:mi>n</mml:mi></mml:math>

Asymptotic normalization coefficients (ANCs) for ${}^{8}{\stackrel{\ensuremath{\rightarrow}}{\mathrm{Li}}}^{7}\mathrm{Li}+n$ have been extracted from the neutron transfer reaction ${}^{13}\mathrm{C}{(}^{7}\mathrm{Li}{,}^{8}\mathrm{Li}{)}^{12}\mathrm{C}$ at 63 MeV. These are related to the ANCs in ${}^{8}{\stackrel{\ensuremath{\rightarrow}}{\mathrm{B}}}^{7}\mathrm{Be}+p$ using charge symmetry. We extract ANCs for ${}^{8}\mathrm{B}$ which are in very good agreement with those inferred from proton transfer and breakup experiments. We have also separated the contributions from the ${p}_{1/2}$ and ${p}_{3/2}$ components in the transfer. We find the astrophysical factor for the ${}^{7}\mathrm{Be}(p,\ensuremath{\gamma}{)}^{8}\mathrm{B}$ reaction to be ${S}_{17}(0)=17.6\ifmmode\pm\else\textpm\fi{}1.7\mathrm{eV}\mathrm{}\mathrm{b}.$ This is the first time that the rate of a direct capture reaction of astrophysical interest has been determined through a measurement of the ANCs in the mirror system.

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