Analogues of the exotic Hoyle state in the $^{12}$C nucleus
Experimental data on inelastic scattering of alpha particles by the $^{11}$B, $^{12}$C, and $^{13}$C nuclei are analyzed using the modified diffraction model and the radii of these nuclei in some "abnormal" excited states are found. It is shown that the (7.65 MeV) Hoyle state in the $^{12}$C nucleus is the base state for a new 0$^{+}_2$--2$^{+}_2$--4$^{+}_2$ rotational band (in addition to the ground-state band), in which the third member is the discovered (13.75 MeV) state. The radii of the $^{12}$C nucleus in the above-mentioned three states are 25--30% larger than its ground state radius. It is found that the radii of the 1/2$^{-}$ (8.86 MeV) state in the $^{13}$C nucleus and the 3/2$^{-}$ (8.56 MeV) state in the $^{11}$B nucleus are close to the radius of the Hoyle state in $^{12}$C and that a similar rotational band is based on the 8.56 MeV state. The above $^{13}$C and $^{11}$B states can be regarded as analogues of the Hoyle state. The prediction of the alpha-condensation model that a similar analogue in $^{11}$B is the 12.56 MeV state with a radius that is comparable with the nuclear radius of uranium was not confirmed.