C12(α,γ)16OE2 cross section in a microscopic four-alpha model

The ${}_{}{}^{12}{}_{}{}^{}\mathrm{C}$(α,γ${)}^{16}$O E2 cross section is calculated in the generator coordinate method where ${}_{}{}^{12}{}_{}{}^{}\mathrm{C}$ is described by three α particles located on the apexes of an equilateral triangle. Distortion effects in the ${}_{}{}^{12}{}_{}{}^{}\mathrm{C}$ wave functions are taken into account, and the ${}_{}{}^{16}{}_{}{}^{}\mathrm{O}$ wave functions are defined by a mixing of α${+}^{12}$C($0^{+}$,$2^{+}$) configurations. The good quantum numbers are restored through angular momentum projection. We compute quadrupole transition probabilities between low-energy ${}_{}{}^{16}{}_{}{}^{}\mathrm{O}$ states, and the ${}_{}{}^{12}{}_{}{}^{}\mathrm{C}$(α,γ${)}^{16}$O E2 cross sections to the ground and $2_1^{+}$ states. At 300 keV, we find an S factor equal to 0.09 MeV b.