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

It is known from the scattering theory that the phase-shift of elastic collision does not provide a unique potential to describe the bound state of the two-particle system. The bound state wave function is the most crucial input for various nuclear processes bearing astrophysical significance. In this paper, we emphasize on the important role of the asymptotic normalization coefficients for description of the $^{13}\rm{N}\longrightarrow {^{13}\rm{C}}+β^++ν_e$ reaction. Using experimental data of the elastic scattering phase-shift of proton and neutron on $^{12}\rm{C}$, we find the asymptotic normalization coefficients which determine the tail of the bound state wave functions. This allows us to modify the wave functions of $^{13}\rm{N}$ and $^{13}\rm{C}$ within the single-particle approach using the inverse scattering theory. These wave functions are used to determine the nuclear matrix element for calculation of the half-life and log{\textit ft} values of $β^+$ decay of $^{13}\rm{N}$. The calculated values of the log{\textit ft} and half-life are smaller than the measured values when the single-particle wave functions are employed. Using overlap functions, instead of single-particle functions, we obtain a better comparison. The overlap function is represented as the product of single-particle function and the corresponding spectroscopic factor.