Algorithm for calculations of asymptotic nuclear coefficients using phase-shift data for charged-particle scattering
Abstract
A new algorithm for the asymptotic nuclear coefficients calculation, which we call the $\mathrm{\ensuremath{\Delta}}$ method, is proved and developed. This method was proposed by Ram\'{\i}rez Su\'arez and Sparenberg (arXiv:1602.04082.) but no proof was given. We apply it to the bound state situated near the channel threshold when the Sommerfeld parameter is quite large within the experimental energy region. As a result, the value of the conventional effective-range function ${K}_{l}({k}^{2})$ is actually defined by the Coulomb term. One of the resulting effects is a wrong description of the energy behavior of the elastic scattering phase shift ${\ensuremath{\delta}}_{l}$ reproduced from the fitted total effective-range function ${K}_{l}({k}^{2})$. This leads to an improper value of the asymptotic normalization coefficient (ANC) value. No such problem arises if we fit only the nuclear term. The difference between the total effective-range function and the Coulomb part at real energies is the same as the nuclear term. Then we can proceed using just this $\mathrm{\ensuremath{\Delta}}$ method to calculate the pole position values and the ANC. We apply it to the vertices $^{4}\mathrm{He}+^{12}\mathrm{C}\ensuremath{\leftrightarrow}^{16}\mathrm{O}$ and $^{3}\mathrm{He}+^{4}\mathrm{He}\ensuremath{\leftrightarrow}^{7}\mathrm{Be}$. The calculated ANCs can be used to find the radiative capture reaction cross sections of the transfers to the $^{16}\mathrm{O}$ bound final states as well as to the $^{7}\mathrm{Be}$.