Single $\alpha$-particle orbits and Bose-Einstein condensation in $^{12}$C

The $\alpha$-bosonic properties such as single-$\alpha$ orbits and occupation numbers in $J^\pi$=$0^+$, $2^+$, $1^-$ and $3^-$ states of $^{12}$C around the $3\alpha$ threshold are investigated with the semi-microscopic $3\alpha$ cluster model. As in other studies, we found that the $0^+_2$ ($2^+_2$) state has dilute-$3\alpha$-condensate-like structure in which the $\alpha$ particle is occupied in the single $S$ ($D$) orbit with about 70% (80%) probability. The radial behaviors of the single-$\alpha$ orbits as well as the occupation numbers are discussed in detail in comparison with those for the $0^+_1$ and $2^+_1$ states together with the $1^-_1$ and $3^-_1$ states.