Energies of molecular structures in ${}^{12}\mathrm{C},{}^{16}\mathrm{O},{}^{20}\mathrm{Ne},{}^{24}\mathrm{Mg},$ and ${}^{32}\mathrm{S}$

The energies of the ${}^{12}\mathrm{C},{}^{16}\mathrm{O},{}^{20}\mathrm{Ne},{}^{24}\mathrm{Mg},$ and ${}^{32}\mathrm{S} 4n$ nuclei have been determined within a generalized liquid drop model and assuming different planar and three-dimensional shapes of the $\alpha$ molecules: linear chain, triangle, square, tetrahedron, pentagon, trigonal bipyramid, square pyramid, hexagon, octahedron, octagon, and cube. The potential barriers governing the entrance and decay channels via $\alpha$ absorption or emission as well as more symmetric binary and ternary reactions have been compared. The rms radii of the linear chains differ from the experimental rms radii of the ground states. The binding energies of the three-dimensional shapes at the contact point are higher than the ones of the planar configurations. The $\alpha$ particle plus A-4 daughter configuration leads always to the lowest potential barrier. The binding energy can be reproduced within the sum of the binding energy of $n \alpha$ particles plus the number of bonds multiplied by 2.4 MeV or by the sum of the binding energies of one $\alpha$ particle and the daughter nucleus plus the Coulomb energy and the proximity energy.

Figure 1

Planar and three-dimensional molecular shapes.

Figure 2

Potential energy of an isosceles triangular $\alpha$ molecule as a function of the angle $\theta$ (deg.) and the root-mean-square radius $Q$.

Figure 3

Potential energies of square and tetrahedral configurations from the contact point as a function of the rms radius.

Figure 4

Potential energy of the $\alpha$ tetrahedron as a function of the angular momentum (in $\hslash$ unit) and rms radius.

Figure 5

Potential energy of the $\alpha$ square as a function of the angular momentum (in $\hslash$ unit) and rms radius.

Figure 6

Potential barriers governing the ${}^{12}\mathrm{C}+{}^4\mathrm{He},{}^8\mathrm{Be}+{}^8\mathrm{Be}$, and linear ${}^6\mathrm{Li}+{}^4\mathrm{He}+{}^6\mathrm{Li}$ nuclear systems versus the distance between the mass centers (at $L=0$).

Figure 7

Potential energies of a pentagon, trigonal bipyramid, and square pyramid from the contact point as a function of the rms radius.

Figure 8

Potential energy of the $\alpha$ bipyramid as a function of the angular momentum (in $\hslash$ unit) and rms radius.

Figure 9

Potential energy of the $\alpha$ square pyramid as a function of the angular momentum (in $\hslash$ unit) and rms radius.

Figure 10

Potential energy of the $\alpha$ pentagon as a function of the angular momentum (in $\hslash$ unit) and rms radius.

Figure 11

Potential barriers governing the ${}^{16}\mathrm{O}+{}^4\mathrm{He},{}^{12}\mathrm{C}+{}^8\mathrm{Be},{}^{10}\mathrm{B}+{}^{10}\mathrm{B}$, and linear ${}^8\mathrm{Be}+{}^4\mathrm{He}+{}^8\mathrm{Be}$ reactions versus the distance between the mass centers (at $L=0$).

Figure 12

Potential energies of a hexagon and an octahedron from the contact point as a function of the rms radius.

Figure 13

Potential barriers governing the ${}^{16}\mathrm{O}+{}^8\mathrm{Be},{}^{12}\mathrm{C}+{}^{12}\mathrm{C}$, linear ${}^8\mathrm{Be}+{}^8\mathrm{Be}+{}^8\mathrm{Be}$, and ${}^{10}\mathrm{B}+{}^4\mathrm{He}+{}^{10}\mathrm{B}$ reactions versus the distance between the mass centers.

Figure 14

Potential energies of an octagonal molecule and a cubic molecule from the contact point as a function of the rms radius.

Figure 15

Potential barriers governing the ${}^{28}\mathrm{Si}+{}^4\mathrm{He},{}^{24}\mathrm{Mg}+{}^8\mathrm{Be},{}^{20}\mathrm{Ne}+{}^{12}\mathrm{C}$, and ${}^{16}\mathrm{O}+{}^{16}\mathrm{O}$ reactions versus the distance between the mass centers.