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 $0_{2}^{+}$ (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 $4_{2}^{+}$ (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.
21.10.Re Collective levels
24.10.Ht Optical and diffraction models
$^1$Department of Physics, Moscow State University, Moscow, 119991 Russia
$^2$Kurchatov Institute National Research Center, Moscow, 123182 Russia
$^3$Universidad Autonomo del Estado de Mexico, Toluca, 5000 Mexico
$^4$University of Jyväskylä, Jyväskylä, Finland