Coulomb energy of $\alpha$-particle aggregates distributed on Archimedean solids

We study the property of $\alpha$ aggregates distributed on Archimedean solids within a microscopic framework, which takes full account of the Pauli principle. We pay special attention to the Coulomb energy for such exotic shapes of nuclei, and we discuss the advantage of $\alpha$ clusters with geometric configurations compared with the uniform density distributions in reducing the repulsive effect. We consider four kinds of configurations of $\alpha$ clusters distributed on Archimedean solids: dual polyhedra composed of a dodecahedron and an icosahedron, an octacontahedron, and two types of truncated icosahedrons, that is, two kinds of Archimedean solids. The latter two are an icosidodecahedron and a fullerene shape. Putting an $\alpha$ cluster on each vertex of the polyhedra, four $\alpha$-cluster aggregates correspond to the following four nuclei: Gd (64 protons), Po (84 protons), Nd (60 protons), and a nucleus with 120 protons, respectively.

Figure 1

Schematic figure for introduced configurations. (a) 30 $\alpha$'s (Nd), (b) 32 $\alpha$'s (Gd), (c) 42 $\alpha$'s, and (d) 60 $\alpha$'s (fullerene). The red and blue balls show $\alpha$ clusters. The red balls represent the $\alpha$ clusters on the vertexes of the icosahedron [(b), (c)], and blue balls come from $\alpha$ clusters on the edges of the icosahedron [(a), (c), (d)]. In (b), the blue balls correspond to the $\alpha$ clusters on the surface of the icosahedron.

Figure 2

Coulomb energy for one $\alpha$ for the Nd isotopes. Thick solid, solid, dotted, dashed, and dash-dotted lines represent the results of $E_c\left(\rho \right)/n_{\alpha }$, $E_{c\left(d\right)}\left(\rho \right)/n_{\alpha }$, $E_{pc}\left(\rho \right)/n_{\alpha }$, $E_{c\left(uS\right)}\left(\rho \right)/n_{\alpha }$, and $E_{c\left(uV\right)}\left(\rho \right)/n_{\alpha }$, respectively. The arrow at $\rho =5.3$ fm gives the nearest-neighbor $\alpha \text{−}\alpha$ distance of 3.3 fm, which is the optimal value in free space.

Figure 3

Coulomb energy for one $\alpha$ for the Gd isotopes. Thick solid, solid, dotted, dashed, and dash-dotted lines represent the results of $E_c\left(\rho \right)/n_{\alpha }$, $E_{c\left(d\right)}\left(\rho \right)/n_{\alpha }$, $E_{pc}\left(\rho \right)/n_{\alpha }$, $E_{c\left(uS\right)}\left(\rho \right)/n_{\alpha }$, and $E_{c\left(uV\right)}\left(\rho \right)/n_{\alpha }$, respectively. The arrow at $\rho =5.2$ fm gives the nearest-neighbor $\alpha \text{−}\alpha$ distance of 3.3 fm, which is the optimal value in free space.

Figure 4

Coulomb energy for one $\alpha$ for the Po isotopes. Thick solid, solid, dotted, dashed, and dash-dotted lines represent the results of $E_c\left(\rho \right)/n_{\alpha }$, $E_{c\left(d\right)}\left(\rho \right)/n_{\alpha }$, $E_{pc}\left(\rho \right)/n_{\alpha }$, $E_{c\left(uS\right)}\left(\rho \right)/n_{\alpha }$, and $E_{c\left(uV\right)}\left(\rho \right)/n_{\alpha }$, respectively. The arrow at $\rho =6.1$ fm gives the nearest-neighbor $\alpha \text{−}\alpha$ distance of 3.3 fm, which is the optimal value in free space.

Figure 5

Coulomb energy for one $\alpha$ for the $Z=120$ (Fullerene) isotopes. Thick solid, solid, dotted, dashed, and dash-dotted lines represent the results of $E_c\left(\rho \right)/n_{\alpha }$, $E_{c\left(d\right)}\left(\rho \right)/n_{\alpha }$, $E_{pc}\left(\rho \right)/n_{\alpha }$, $E_{c\left(uS\right)}\left(\rho \right)/n_{\alpha }$, and $E_{c\left(uV\right)}\left(\rho \right)/n_{\alpha }$, respectively. The arrow at $\rho =8.2$ fm gives the nearest-neighbor $\alpha \text{−}\alpha$ distance of 3.3 fm, which is the optimal value in free space.