Properties of ${}^{12}$C resonances determined from the ${}^{10}$B(${}^3$He,$p\alpha \alpha \alpha$) and ${}^{11}$B(${}^3$He,$d\alpha \alpha \alpha$) reactions studied in complete kinematics

We have used the ${}^{10}$B(${}^3$He$,p\alpha$$\alpha$$\alpha$) reaction at 4.9 MeV and the ${}^{11}$B(${}^3$He$,d\alpha$$\alpha$$\alpha$) reaction at 8.5 MeV to determine the energy, width, and decay mechanism of resonances in ${}^{12}$C above the triple-$\alpha$ threshold up to 21 MeV in excitation energy. Comparison with various models has allowed us to estimate the degree of clustering of the states studied.

Figure 1

Multiplicity-four data from the ${}^{10}$B(${}^3$He$,p\alpha \alpha \alpha$) reaction. Individual $\alpha$ energies are plotted against proton energies in the c.m. frame. Excited states in ${}^{12}$C appear as vertical bands of increased intensity in the two-dimensional spectrum, shown as peaks in the projections above the $E_{{\alpha }_i}$ vs $E_p$ plot figure. The data have been split into reactions that proceed via the ground state in ${}^8$Be (a) and the rest (b), as explained in the text. The contribution from the ground state of ${}^5$Li is circled in black in the lower part of (a). Similarly, the contribution from the narrow 2.36 MeV state in ${}^9$B is circled in grey (pink) in (b). Both of these contributions have been removed before performing the fits of the resonances. There are clearly structures from broader states visible in both plots which cannot be fully removed.

Figure 2

Same as Fig. 1, but for the ${}^{11}$B(${}^3$He$,d\alpha \alpha \alpha$) reaction. It is clear from the projections that the ${}^{12}$C excitation energy spectrum is cleaner than that obtained with the ${}^{10}$B target. The contribution from the narrow 2.19 MeV state in ${}^6$Li is circled in black. Its contribution to the ${}^{12}$C excitation-energy spectrum was removed before performing the fits.

Figure 3

Ground-state branching ratios vs $r_0$. The green-shaded bands indicate a 1$\sigma$ uncertainty obtained by adding the systematic and statistical uncertainties in quadrature. A value of $r_0=1.41$ fm was used to determine the branching ratios.