Structure and Rotations of the Hoyle State

The excited state of the ${}^{12}\mathrm{C}$ nucleus known as the “Hoyle state” constitutes one of the most interesting, difficult, and timely challenges in nuclear physics, as it plays a key role in the production of carbon via fusion of three alpha particles in red giant stars. In this Letter, we present ab initio lattice calculations which unravel the structure of the Hoyle state, along with evidence for a low-lying spin-2 rotational excitation. For the ${}^{12}\mathrm{C}$ ground state and the first excited spin-2 state, we find a compact triangular configuration of alpha clusters. For the Hoyle state and the second excited spin-2 state, we find a “bent-arm” or obtuse triangular configuration of alpha clusters. We also calculate the electromagnetic transition rates between the low-lying states of ${}^{12}\mathrm{C}$.

Figure 1

Lattice results for the ${}^{12}\mathrm{C}$ spectrum at leading order (LO). Panel I shows the results using initial states $A$, $B$, and $\Delta$, each of which approaches the ground state energy. Panel II shows the results using initial states $C$, $D$, and $\Lambda$. These trace out an intermediate plateau at an energy $\simeq 7\mathrm{MeV}$ above the ground state.

Figure 2

Illustration of the initial state $\Delta$. There are 12 equivalent orientations of this compact triangular configuration.

Figure 3

Illustration of the initial state $\Lambda$. There are 24 equivalent orientations of this bent-arm or obtuse triangular configuration.