High-precision proton angular distribution measurements of ${}^{12}\mathrm{C}(p,p^{\prime })$ for the determination of the $E0$ decay branching ratio of the Hoyle state

Background: In stellar environments, carbon is produced exclusively via the $3\alpha$ process, where three $\alpha$ 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 ${}^{12}\mathrm{C}(p,p^{\prime })$ 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 $\left(0_2^{+}\right)$ and $2_1^{+}$ state in ${}^{12}\mathrm{C}$ in ${}^{12}\mathrm{C}(p,p^{\prime })$ 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 ${}^{12}\mathrm{C}(p,p^{\prime })$ reactions with a silicon detector array covering ${22}^{\circ }<\theta <{158}^{\circ }$ in 14 small energy steps between 10.20 and 10.70 MeV with a thin $(60\mu \mathrm{g}$/${\mathrm{cm}}^2)$ ${}^{\mathrm{nat}}\mathrm{C}$ 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 ${}^{12}\mathrm{C}(p,p^{\prime })$ 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 ${}^{13}\mathrm{N}$.

Figure 1

The decay of ${}^{12}\mathrm{C}$ from the Hoyle state primarily occurs via sequential $\alpha$ decay, into ${}^8\mathrm{Be}(\to \alpha +\alpha )+\alpha$. A very small fraction of $\alpha$ decays proceed via direct $3\alpha$ decay ($<0.019\%–0.043\%$) [5, 6, 7, 8]. Stable carbon is produced when ${}^{12}\mathrm{C}$ decays electromagnetically ($0.04\%–0.06\%$ of decays) via a cascade of two $E2$ transitions, or one $E0$ transition [9, 10]. The branching ratios indicated in the figure arise from Ref. [10].

Figure 2

(a) Laboratory total energy vs ${\theta }_{\mathrm{lab}}$ spectrum for particles detected at 10.30 MeV in the DSSD array. (b) Total energy spectrum in a ${76}^{\circ }<{\theta }_{\mathrm{lab}}<{85}^{\circ }$ slice, corresponding to a single detector arc in the BALiN array. The features of these spectra are discussed in the text.

Figure 3

Differential cross sections (blue squares) in the laboratory frame for ${}^{12}\mathrm{C}(p,p^{\prime })$ populating the $0_2^{+}$ state at 10.30 MeV (a) and 10.48 MeV (b), and the $2_1^{+}$ state at at the same energies (c),(d). Other energies are shown in Appendix pp2. Error bars, smaller than the points, are purely statistical. Magenta circles indicate cross sections from Ref. [16] which have been digitized and converted to the laboratory frame. In panel (b), the cross section from Ref. [16] at $25.{54}^{\circ }$ (indicated by the magenta arrow) lies off scale at 113(4) mb/deg (discussed in the text). Light blue curves represent $1\sigma$ confidence intervals of fifth-order Legendre polynomial fits to the differential cross sections $d\sigma /d\mathrm{\Omega }\left(\theta \right)$.

Figure 4

Total cross sections for (a) the $0_2^{+}$ and (b) $2_1^{+}$ states between $E_{lab}=10.20$ and 10.70 MeV (blue points). The error bars arise from the $1\sigma$ confidence limits of the fits shown in Figs. 3, 5, and 6. Cross sections for the $2_1^{+}$ state from Ref. [20] are shown in panel (b) by the orange squares. $N_p\left(2_1^{+}\right)/N_p\left(0_2^{+}\right)$ ratios are shown in (c), zoomed around 10.38–10.50 MeV in panel (d). The value of Ref. [15] is shown by the magenta square. The ratio from the 1 mg/${\mathrm{cm}}^2$ target measurement (oriented at ${45}^{\circ }$) is shown by the purple diamond. Taking the weighted average of the thin target (blue) points over the energy loss expected in a 1 mg/${\mathrm{cm}}^2$ target oriented at ${45}^{\circ }$ yields the green triangle. For the ratios extracted for thick targets, the energy range covered is indicated by the horizontal bar. The agreement of the “equivalent target” with the actual thick target measurement is excellent.

Figure 5

Angular distributions for ${}^{12}\mathrm{C}(p,p^{\prime })$ populating the $0_2^{+}$ state. Error bars (smaller than the points) are purely statistical. Light blue curves represent fifth-order Legendre polynomial fits to $d\sigma /d\mathrm{\Omega }\left(\theta \right)$ to enable extrapolation beyond the detection region. The width of the light blue curves show the $1\sigma$ confidence interval of the fit. The results for $E_{\mathrm{Beam}}=10.30$ and 10.48 MeV are shown in Figs. 3 and 3.

Figure 6

Angular distributions for ${}^{12}\mathrm{C}(p,p^{\prime })$ populating the $2_1^{+}$ state. Error bars (smaller than the points) are purely statistical. Light blue curves represent fifth order Legendre polynomial fits to $d\sigma /d\mathrm{\Omega }\left(\theta \right)$ to enable extrapolation beyond the detection region. The width of the light blue curves show the $1\sigma$ confidence interval of the fit. The results for $E_{\mathrm{Beam}}=10.30$ and 10.48 MeV are shown in Fig. 3 and 3.

Figure 7

Elastic scattering angular distributions for ${}^{\mathrm{nat}}\mathrm{C}(p,p)$ reactions. Error bars (smaller than the points) are purely statistical. These may also be regarded as ${}^{12}\mathrm{C}(p,p)$ cross sections for which there is an additional systematic error of up to $1.06\%$ at each angle due to the contribution of ${}^{13}\mathrm{C}(p,p)$ that could not be separated.