The $\alpha {+}^{12}\mathrm{C}$ $d$ phase shift and the ${}^{12}\mathrm{C}(\alpha ,\gamma {)}^{16}\mathrm{O}$ $E2\mathrm{}$ $S$ factor are analyzed in the $R$-matrix theory. A new method is proposed to link experimental and calculated parameters within the $R$-matrix formalism. We show that interference patterns in the phase shift near 3 MeV are very helpful to constrain the parameter set. Starting from this analysis, we fit the ${}^{12}\mathrm{C}(\alpha ,\gamma {)}^{16}\mathrm{O}$ $E2\mathrm{}$ $S$ factor, and find low-energy values larger than currently accepted. Different interference patterns are found, but do not appear to affect the astrophysical energies. Consequently, direct measurements at low energies are necessary for astrophysical applications.

• Received 8 September 1999

DOI:https://doi.org/10.1103/PhysRevC.61.064611