Conversion between the formal and observed parameters in $R$-matrix theory

The conversion between the formal and observed parameters is obtained from the phase equivalence between the $R$-matrix and Breit-Wigner formula, and it is applied to a study of the low-energy $E$1 $S$-factor of ${}^{12}$C($\alpha$,${\gamma }_0$)${}^{16}$O. As an example, weak interference between 1${}^{-}$ states in ${}^{16}$O is discussed. As well as the ordinary $R$-matrix method, the present method calculates the $S$-matrix and cross sections, independent of the boundary condition. Therefore, the $E$1 $S$-factor at $E_{c\mathrm{.}m\mathrm{.}}=300$ keV is found to be reduced from the current evaluation.