Dilute multi-<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mi>α</mml:mi></mml:mrow></mml:math>cluster states in nuclei

Dilute multi-$\ensuremath{\alpha}$ cluster condensed states with spherical and axially deformed shapes are studied with the Gross-Pitaevskii equation and Hill-Wheeler equation, where the $\ensuremath{\alpha}$ cluster is treated as a structureless boson. Applications to self-conjugate $4N$ nuclei show that the dilute $N\ensuremath{\alpha}$ states of $^{12}\mathrm{C}$ to $^{40}\mathrm{Ca}$ with ${J}^{\ensuremath{\pi}}={0}^{+}$ appear in the energy region from threshold up to about $20\phantom{\rule{0.3em}{0ex}}\text{MeV}$, and the critical number of $\ensuremath{\alpha}$ bosons that the dilute $N\ensuremath{\alpha}$ system can sustain as a self-bound nucleus is estimated roughly to be ${N}_{cr}\ensuremath{\sim}10$. We discuss the characteristics of the dilute $N\ensuremath{\alpha}$ states with emphasis on the $N$ dependence of their energies and rms radii.