Dilute Multi Alpha Cluster States in Nuclei

Dilute multi $\alpha$ cluster condensed states with spherical and axially deformed shapes are studied with the Gross-Pitaevskii equation and Hill-Wheeler equation, where the $\alpha$ cluster is treated as a structureless boson. Applications to self-conjugate $4N$ nuclei show that the dilute $N\alpha$ states of $^{12}$C to $^{40}$Ca with $J^\pi=0^+$ appear in the energy region from threshold up to about 20 MeV, and the critical number of $\alpha$ bosons that the dilute $N\alpha$ system can sustain as a self-bound nucleus is estimated roughly to be $N_{cr}\sim10$. We discuss the characteristics of the dilute $N\alpha$ states with emphasis on the $N$ dependence of their energies and rms radii.