Study of spatial structures in $$\alpha $$ α -cluster nuclei

Calculations of the probability densities and energies of the ground states for $$\alpha $$ -cluster nuclei $${}^{12}$$ C (3 $$\alpha $$ ), $${}^{16}$$ O (4 $$\alpha $$ ), $${}^{20}$$ Ne ( $$5\alpha $$ ), $${}^{24}$$ Mg ( $$6\alpha $$ ), $${}^{28}$$ Si ( $$7\alpha $$ ) and for nuclear molecules $${}^{9}$$ Be ( $$2\alpha +n$$ ), $${}^{10}$$ Be ( $$2\alpha +2n$$ ), $${}^{10}$$ B ( $$2\alpha +n+p$$ ), $${}^{10}$$ C ( $$2\alpha +2p$$ ), $${}^{11}$$ B ( $$2\alpha +2n+p$$ ), $${}^{11}$$ C ( $$2\alpha +n+2p$$ ) were performed by the Feynman Path Integral method using parallel computing based on NVIDIA CUDA technology. In addition, $${}^{12}$$ C (3 $$\alpha $$ ) was considered in three-body hyperspherical harmonics formalism with a new effective method of solving hyperradial equation using cubic spline interpolation. The spatial structures of the ground states of $${}^{12}$$ C and $${}^{16}$$ O correspond to simple geometric shapes, a regular triangle and a tetrahedron, respectively, with wide $$\alpha $$ -cluster clouds in their vertices. The $${}^{24}$$ Mg nucleus has the shape of a pair of equal regular triangles with wide $$\alpha $$ -cluster clouds in their vertices as well. The nuclei $${}^{9}$$ Be, $${}^{10}$$ Be, $${}^{10}$$ B, $${}^{10}$$ C, $${}^{11}$$ B, $${}^{11}$$ C have molecular structure with valence neutrons and/or protons between two $$\alpha $$ -clusters.