Evidence of the $3\alpha$-particle condensate character of the Hoyle state (the $0_2^{+}$ state at $7.65$ MeV in ${}^{12}\mathrm{C}$) implies not only an enhanced radius of ${}^{12}\mathrm{C}$ in this state, which was established by many theoretical calculations and confirmed by the recent diffraction model analysis, but also zero relative angular momenta between clusters. We performed coupled-channels model calculations of the angular distributions of $\alpha +{}^{12}$C elastic and inelastic (to the $4.44$-MeV $2$${}^{+}$, $7.65$-MeV $0$${}_2^{+}$, and $9.65$-MeV $3_1^{-}$ states) scattering at $110$ MeV and found the ratio of the empirical spectroscopic factors $S(L)$. As the differential cross sections of these reactions are characterized by pronounced enhancement and strong oscillations at large angles, we assumed a potential scattering in the forward hemisphere and the direct transfer of a ${}^8\mathrm{Be}$ cluster at ${\theta }_{\mathrm{c}.\mathrm{m}.}>{90}^{°}$ and took into account the direct transfer of ${}^8\mathrm{Be}$ in the ground state and in the first excited $2^{+}$ and $4^{+}$ states. We found that the cluster configuration with $L=0$ dominates in the $0_2^{+}$ state, being more than three times larger than that in the ground state. This result provides additional evidence of the condensed structure of the Hoyle state in ${}^{12}\mathrm{C}$ with a dominance of zero relative angular momentum. The negative-parity $3_1^{-}$ excited state in ${}^{12}\mathrm{C}$ observed above the $3\alpha$ threshold is also considered to have the $3\alpha$-cluster structure. The present calculations described well the structure of the large-angle cross section on this state. We found a positive interference for all allowed $\alpha +{}^8$Be configurations with a dominance of the $p$-orbital $(69\%)\alpha +{}^8$Be motion and confirmed the exotic, but hardly a condensed, structure of this state.

• Received 10 September 2010

DOI:https://doi.org/10.1103/PhysRevC.82.054618