Multicluster study of the<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:msup><mml:mrow/><mml:mrow><mml:mn>12</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mi mathvariant="normal">C</mml:mi><mml:mo>+</mml:mo><mml:mi>n</mml:mi></mml:math>and<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:msup><mml:mrow/><mml:mrow><mml:mn>12</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mi mathvariant="normal">C</mml:mi><mml:mo>+</mml:mo><mml:mi>p</mml:mi></mml:math>systems

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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