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The Polyhedral Structure of Nuclear Shells

There is overwhelming evidence that the neutrons and protons which compose nuclei are arranged separately in shells. This was initially discerned by the physicists Maria Goeppert-Mayer and Hans Jensen from the statistics on the numbers of stable isotopes and isotones of the elements. Goeppert-Mayer and Jensen were awarded the Nebel Prize in physics for 1963. Goeppert-Mayer and Jensen characterized the numbers which corresponded to filled shells as magic numbers. The magic numbers Goeppert-Mayer and Jensen identified were {2, 8, 20, 28, 50, 82, 126}. These numbers mean that the capacities of the shells were {2, 6, 12, 8, 22, 32, 44}.

The statistics on the binding energies of nuclides provide a sharper means of identifying the transition points corresponding to filled shells. Binding energy is the mass deficit of nuclides expressed in energy terms. The mass deficit of a nuclide is the difference between the mass of the neutrons and protons which make it up and its mass. The incremental binding energies are computed as the change in binding energy as a result of adding one more neutron or one more proton to the nuclide. The incremental binding energies shows increases that fluctuate on the basis of whether an additional nucleon forms or does not form a pairing with a nucleon of its type. The incremental binding energy shows a sharp drop after a shell is formed and the next nucleon goes into a different shell. The incremental binding energy data identifies the magic numbers of Goeppert-Mayer and Jensen, but the data shows that 6 and 14 are also magic numbers.

An algorithm was developed elsewhere which generates all of the magic numbers except 8 and 20. These appear to be special cases. According to that algorithm the capacities of the shells are {2, 4, 8, 14, 22, 32, 44} and beyond the existing data to 58. However that algorithm indicates that for each shell there are two arrangements that do not fit in with the structure of the rest of the shell. According to that insight from the algorithm the shell capacities would be {2, 6, 12, 20, 30, 42}. Note that these numbers are the product of successive integers; i.e., {1*2, 2*3, 3*4, 4*5, 5*6, 6*7}.

It is expected that the nucleons in the shells would have some polyhedral structure just as do the carbon atoms in the buckminsterfullerenes. In the fullerenes the carbon atoms are arrayed in hexagons and pentagons. In the case of the Helium 4 (alpha particle) nuclide the structure seems to be that of a rotating tetrahedron. The rotation is necessary to maintain a balance of forces among the nucleons. Also this is a structure involving the neutrons and protons together. The nuclear shells generally refer to a shells of neutrons and shells of protons separately. There is experimental evidence that the deuteron (Hydrogen 2) has an empty center, as would be expected if the neutron and proton are rotating about their center of mass.

The structure of the Platonic and Archimedean Polyhedra

The numbers and type of faces, the number of edges and the numbers and degrees of the vertices of the Platonic and Archimedean polyhedra are tabulated below.

v k n fn m fm p fp e Name
4 3 3 4 6 tetrahedron
6 4 3 8 12 octahedron
8 3 4 6 12 cube
12 3 3 4 6 4 18 truncated
12 4 3 8 4 6 24 cuboctahedron
12 5 3 20 30 icosahedron
20 3 5 12 30 dodecahedron
24 3 3 8 8 6 36 truncated cube
24 3 4 6 6 8 36 truncated
24 4 3 8 4 18 48 rhombicuboctahedron
24 5 3 32 4 6 60 snub cuboctahedron
30 4 3 20 5 12 60 icosidodecahedron
48 3 4 12 6 8 8 6 72 truncated
60 3 3 20 10 12 90 truncated
60 3 5 12 6 20 90 truncated icosahedron
60 4 3 20 4 30 5 12 120 rhombicosidodecahedron
60 5 3 80 5 12 150 snub
120 3 4 30 6 20 10 12 180 truncated


The information of primary interest now is the number of vertices. These are {4, 6, 8, 12, 20, 24, 30, 48, 60, 120}. There is a major overlap with the revised set of shell capacities {6, 12, 20, 30, 42, 56}. The match for the first four is promising. There are other polyhedra besides the Platonic and Archimedean. For example there are the polygonal prisms. A polyhedron with a seven sided prism consisting of 6 bands of vertices has 42 vertices.

At appears that there is an indication that one of the shells has an octahedral structure, but in order to maintain a structure the shell must be rotating to balance the force of attraction between the nucleons. Therefore the shell structure can have no nucleons on the spin axis. So the spin axis could not coincide with the axis of symmetry of the octahedron. That is entirely possible but an alternate polyhedral structure is possible; i.e. that of a triangular prism.

The structure of the shell with a capacity of 12 could be an icosahedron, a cuboctahedron or a truncated tetrahedron. It also could be that of a hexagonal prism or twisted prism. The one with a capacity of 20 could have a dodecahedral structure, but also that of four pentagonal rings. And the one with a capacity of 30 could have the structure of an icosidodecahedron or five hexagonal rings or six pentagonal rings. There could be numerous prismatic or ring structures and no reason for the special numbers {6, 20, 30}. It is the polyhedra that give a justification for those numbers.

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