States of the $^{12}$C Nucleus in the Toroidal Configuration

The $^{12}$C nucleus with $N$=6 and $Z$=6 is a doubly closed-shell nucleus in a toroidal potential. In the description of the ground state and the Hoyle state of $^{12}$C in the resonating group method or the generator coordinate method, a superposition of the orientations of Wheeler's triangular cluster on the cluster plane would naturally generate an intrinsic toroidal density. A toroidal state also has a probability amplitude to overlap with a 3-alpha cluster, which is the dominant decay mode for the Hoyle state. For these reasons, we study a toroidal description of the states of $^{12}$C in the toroidal configuration both phenomenologically and microscopically. A toroidal $^{12}$C nucleus distinguishes itself by toroidal particle-hole multiplet excitations between one toroidal single-particle shell to another. From such a signature and experimental data, we find phenomenologically that the Hoyle state and many of its higher excited states may be tentatively attributed to those of the $^{12}$C nucleus in a toroidal configuration. We then study the $^{12}$C system from a microscopic mean-field approximation using variational wave functions. We find that the equidensity surfaces of the $^{12}$C ground state exhibit a dense toroidal core immersed in lower-density oblate spheroids in the surface region. Furthermore, there are prominent toroidal features of the equidensity surfaces for the state at the Hoyle excitation energy, at which previous cluster model calculations indicate the presence of a 3-alpha cluster state. A toroidal coexistence model therefore may emerge to suggest the possibility that the physical Hoyle state may have probability amplitudes to be in the toroidal configuration and the 3-alpha cluster configuration.