Certain aspects of the recently proposed antisymmetrized $\alpha$-particle product state wave function, or THSR (Tohsaki-Horiuchi-Schuck-Röpke) $\alpha$-cluster wave function, for the description of the ground state in ${}^8\mathrm{Be}$, the Hoyle state in ${}^{12}\mathrm{C}$, and analogous states in heavier nuclei are elaborated in detail. For instance, the influence of antisymmetrization in the Hoyle state on the bosonic character of the $\alpha$ particles is studied carefully. It is shown to be weak. Bosonic aspects in Hoyle and similar states in other self-conjugate nuclei are, therefore, predominant. Another issue is the de Broglie wavelength of $\alpha$ particles in the Hoyle state, which is shown to be much larger than the inter-$\alpha$ distance. It is pointed out that the bosonic features of low-density $\alpha$ gas states have measurable consequences, one of which, enhanced multi-$\alpha$ decay properties, has likely already been detected. Consistent with experiment, the width of the proposed analog to the Hoyle state in ${}^{16}\mathrm{O}$ at the excitation energy of $E_x=15.1$ MeV is estimated to be very small ($34$ keV), lending credit to the existence of heavier Hoyle-like states. The intrinsic single-boson density matrix of a self-bound Bose system can, under physically desirable boundary conditions, be defined unambiguously. One eigenvalue then separates out, being close to the number of $\alpha$ particles in the system. Differences between Brink and THSR $\alpha$-cluster wave functions are worked out. No cluster model of the Brink type can describe the Hoyle state with a single configuration. On the contrary, many superpositions of the Brink type are necessary, implying delocalization toward an $\alpha$-product state. It is shown that single $\alpha$-particle orbits in condensates of different nuclei are almost the same. It is thus argued that $\alpha$-particle (quartet) antisymmetrized product states of the THSR type are a very promising novel and useful concept in nuclear physics.

• Received 7 February 2009

DOI:https://doi.org/10.1103/PhysRevC.80.064326