Angular distributions of ${}^{12}\mathrm{C}(\alpha ,\alpha {)}^{12}\mathrm{C}$ have been measured for $E_{\alpha }=2.6\mathrm{}–8.2\mathrm{MeV}$, at angles from $24°$ to $166°$, yielding 12 864 data points. R-matrix analysis of the ratios of elastic scattering yields a reduced width amplitude of ${\gamma }_{12}=0.47\mathrm{}\pm 0.06{\mathrm{MeV}}^{1/2}$ for the $E_x=6.917\mathrm{}\mathrm{MeV}\left(2^{+}\right)$ state in ${}^{16}\mathrm{O}(a=5.5\mathrm{}\mathrm{fm})$. The dependence of the ${\chi }^2$ surface on the interaction radius $a$ has been investigated and a deep minimum is found at ${a=5.42\mathrm{}}_{-0.27\mathrm{}}^{+0.16\mathrm{}}\mathrm{fm}$. Using this value of ${\gamma }_{12}$, radiative $\alpha$ capture and ${}^{16}\mathrm{N}$ $\beta$-delayed $\alpha$-decay data, the S factor is calculated at $E_{\mathrm{c}.\mathrm{m}.\mathrm{}}=300\mathrm{}\mathrm{keV}$ to be $S_{E2\mathrm{}}\left(300\right){=53\mathrm{}}_{-18\mathrm{}}^{+13\mathrm{}}\mathrm{keV}\mathrm{b}$ for destructive interference between the subthreshold resonance tail and the ground state $E2\mathrm{}$ direct capture.

• Received 6 September 2001

DOI:https://doi.org/10.1103/PhysRevLett.88.072501