Three-body model of the $^{12}\rm C$ nucleus with distortion of $\alpha$-clusters

The Faddeev equations in configuration space are used to study $^{12}$C nucleus considered as 3$\alpha$-cluster system. The model includes a phenomenological (Ali-Bodmer) pair potential, a three-body potential, and takes into account the Coulomb interaction. The range parameter of the three-body potential is fixed by adjusting the position of the diffraction minimum of the $^{12}$C elastic form factor. It is shown that the model must be supplemented by assumption of a distortion in the charge density of an $\alpha$ cluster inside the $^{12}$C nucleus. The model allows to reproduce well observed characteristics of the low-lying $0^{+}$ levels for $^{12}$C.