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We use the generator coordinate method to study the ${}^{12}$C($n,\ensuremath{\gamma}{)}^{13}$C and ${}^{12}$C($p,\ensuremath{\gamma}{)}^{13}$N reactions, as well as the ${}^{13}$C and ${}^{13}$N spectroscopy. The ${}^{12}$C wave functions are defined by three $\ensuremath{\alpha}$ particles in a regular triangle of size ${R}_{C}.$ Different configurations are considered, in order to analyze clustering effects. It is shown that spectroscopic properties of ${}^{13}$C and ${}^{13}$N are sensitive to the ${}^{12}$C wave function; reasonable agreement with experiment is found with ${R}_{C}$ values minimizing the ${}^{12}$C binding energy. The present study supports the suggestion of a halo structure for the 1/2${}^{+}$ excited state in ${}^{13}$C. The neutron and proton capture cross sections are in good agreement with experiment. Finally, we analyze distortion effects in the ${}^{12}\mathrm{C}+n$ wave functions.

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