High-precision proton angular distribution measurements of C 12 ( p , p ′ ) for the determination of the E 0 decay branching ratio of the Hoyle state

Background: In stellar environments, carbon is produced exclusively via the $3\ensuremath{\alpha}$ process, where three $\ensuremath{\alpha}$ particles fuse to form $^{12}\mathrm{C}$ in the excited Hoyle state, which can then decay to the ground state. The rate of carbon production in stars depends on the radiative width of the Hoyle state. While not directly measurable, the radiative width can be deduced by combining three separately measured quantities, one of which is the $E0$ decay branching ratio. The $E0$ branching ratio can be measured by exciting the Hoyle state in the $^{12}\mathrm{C}(p,{p}^{\ensuremath{'}})$ reaction and measuring the pair decay of both the Hoyle state and the first ${2}^{+}$ state of $^{12}\mathrm{C}$.Purpose: We aim to reduce the uncertainties in the carbon production rate in the universe by measuring a set of proton angular distributions for the population of the Hoyle state $({0}_{2}^{+})$ and ${2}_{1}^{+}$ state in $^{12}\mathrm{C}$ in $^{12}\mathrm{C}(p,{p}^{\ensuremath{'}})$ reactions between 10.20 and 10.70 MeV, used in the determination of the $E0$ branching ratio of the Hoyle state.Method: Proton angular distributions populating the ground, first ${2}^{+}$, and the Hoyle states in $^{12}\mathrm{C}$ were measured in $^{12}\mathrm{C}(p,{p}^{\ensuremath{'}})$ reactions with a silicon detector array covering ${22}^{\ensuremath{\circ}}l\ensuremath{\theta}l{158}^{\ensuremath{\circ}}$ in 14 small energy steps between 10.20 and 10.70 MeV with a thin $(60\phantom{\rule{4pt}{0ex}}\ensuremath{\mu}\mathrm{g}$/${\mathrm{cm}}^{2})$ ${}^{\mathrm{nat}}\mathrm{C}$ target.Results: Total cross sections for each state were extracted and the population ratio between the ${2}_{1}^{+}$ and Hoyle state determined at each energy step. By appropriately averaging these cross sections and taking xtheir ratio, the equivalent population ratio can be extracted applicable for any thick $^{12}\mathrm{C}$ target that may be used in pair-conversion measurements. This equivalent ratio agreed with a direct measurement performed with a thick target.Conclusions: We present a general data set of high-precision $^{12}\mathrm{C}(p,{p}^{\ensuremath{'}})$ cross sections that make uncertainties resulting from the population of the ${2}_{1}^{+}$ and ${0}_{2}^{+}$ states by proton inelastic scattering negligible for any future measurements of the $E0$ branching ratio in $^{12}\mathrm{C}$. Implications for future measurements are discussed, as well as possible applications of this data set for investigating cluster structures in $^{13}\mathrm{N}$.