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It is known from the scattering theory that the phase-shift of elasticcollision does not provide a unique potential to describe the bound state ofthe two-particle system. The bound state wave function is the most crucialinput for various nuclear processes bearing astrophysical significance. In thispaper, we emphasize on the important role of the asymptotic normalizationcoefficients for description of the $^{13}\rm{N}\longrightarrow{^{13}\rm{C}}+\beta^++\nu_e$ reaction. Using experimental data of the elasticscattering phase-shift of proton and neutron on $^{12}\rm{C}$, we find theasymptotic normalization coefficients which determine the tail of the boundstate wave functions. This allows us to modify the wave functions of$^{13}\rm{N}$ and $^{13}\rm{C}$ within the single-particle approach using theinverse scattering theory. These wave functions are used to determine thenuclear matrix element for calculation of the half-life and log{\textit ft}values of $\beta^+$ decay of $^{13}\rm{N}$. The calculated values of thelog{\textit ft} and half-life are smaller than the measured values when thesingle-particle wave functions are employed. Using overlap functions, insteadof single-particle functions, we obtain a better comparison. The overlapfunction is represented as the product of single-particle function and thecorresponding spectroscopic factor.

Reexamination of the $\beta^+$ decay process of $^{13}\rm{N}$ nucleus