Radiative cascade transitions and the12C(α,γ)16OE2cross section to the ground state of16O

The $\alpha$-particle width of the 6.917 MeV, $J^{\pi }{=2\mathrm{}}^{+}$ state in ${}^{16}\mathrm{O}$ is the most uncertain value in the determination of the $E2\mathrm{}$ radiative ground state transition of the astrophysically important ${}^{12}\mathrm{C}(\alpha ,\gamma {)}^{16}\mathrm{O}$ reaction at energies relevant to stellar helium burning. As discussed previously in literature it is possible in principle to derive a precise value of the $\alpha$ width of the 6.917 MeV state by measuring the $E2\mathrm{}$ direct capture into this state. Here the theory based on the R-matrix formalism connecting the cascade direct capture and the $E2\mathrm{}$ ground state transition is described in more complete form than hitherto and resulting direct capture cross sections are estimated. In addition, conditional fits are made to existing data and the resulting value of the $\alpha$-particle width is discussed. From these considerations the cascade cross sections are found to be of the order of 10 nb in an extended energy range proposed to be measured. Angular distributions for the primary $\gamma$ transition are also given. Conclusions for future experiments are then drawn. The branching ratio of the $7.\overrightarrow{1}6.9\mathrm{MeV}\gamma$ transition should also be measured.