Proton halo in the $^{13}\mathrm{N}$ nucleus

We demonstrate that the radii of excited nuclear states can be estimated using the $(^3\mathrm{He}, t)$ charge-exchange reaction and relying on the modified diffraction model. The radius of the $\mathrm{N}$ excited state with an excitation energy of $E^*=2.73$ MeV, which lies in a continuous spectrum, is determined. The radius of this state proves to be close to that of the mirror $3.09$-MeV state of the $^{13}\mathrm{C}$ nucleus, which possesses a neutron halo but lies in a discrete spectrum. Thereby, we demonstrate that the $2.37$-MeV state of the $^{13}\mathrm{N}$ nucleus has a proton halo. The analysis is based on published measurements of differential cross sections for relevant reactions.