The evidence of scattering of electrons by nuclei is that generally nuclei are approximately spherical. The deviation from sphericality is mmeasured by the magnetic quadrupole moment. The big question is whether that sphericality represents the static shape of the nuclei or whether it is a dynamic time-averaged shape.

The Copenhagen Interpretation (CI) of the pattern of the electrons in an atom is that the electrons have no position and trajectories. According to the CI, due to the Uncertainty Principle, the electrons of an atome exist only as probabiity density distributions. The case is made here is that periodic motion of the components of nuclei create the appearance of indeterminacy with a spheroidal shape.

The Nature of the Nucleonic
Substructures of in Nuclei

The mass of a nucleus containing n neutrons and p protons is less that the combined masses of n neutrons and p protons. The difference is called the mass deficit of the nuclide. When the mass deficit is expressed in energy units via the Einstein formula E=mc² the value is called the binding energy of the nuclide.

If BE(n, p) is the binding energy of a nuclide with n neutrons and p protons then the incremental binding energy IBEN of a neutron is

IBEN(n, p) = BE(n, p) − BE(n-1, p)

The incremental binding energy of a proton, IBEP, is

IBEN(n, p) = BE(n, p) − BE(n, p-1)

These two incremental binding energies provide overwhelming evidence for the following facts.

• Without a doubt, two neutrons whenever possible form a neutron-neutron spin pair. This is evidenced by the sawtooth character of the relationship between the incremental binding energies of neutrons and the number of neutrons in the nuclide. For example,
  

  Without a doubt, a neutron forms a neutron-neutron spin pair with only one neutron.

• Without a doubt, two protons whenever possible form a proton-proton spin pair. This is evidenced by the sawtooth character of the relationship between the incremental binding energies of protons and the number of protons in the nuclide. For example,
  

  Without a doubt, a proton forms a proton-proton spin pair with only one proton.

• Without a doubt, a neutron and a proton whenever possible form a neutron-proton spin pair. This is evidenced by what happens to the incremental binding energy of a neutron when the number of neutrons exceeds the number of protons.
  

  When the number of neutrons is below the number of protons the addition of another neutron creates a neutron-proton spin pair. It also creates binding energy due to the interaction through the strong force of the additional neutron with the neutrons and protons already in the nuclide. If the number of neutrons is odd then another neutron will create a neutron-neutron spin pair.

  The same applies to the incremental binding energy of a proton when the number of protons exceeds the number of neutrons.

  
• Without a doubt neutrons and protons are arranged in shells. This is evidenced by the sharp drop in incremental binding energy when the number of neutrons or the number of protons exceeds certain numbers, commonly called magic numbers. The conventional magic numbers are {2, 8, 20, 28, 50, 82, 126}. The case is made elsewhere that 6 and 14 are also magic numbers and that 8 and 20 are in the nature of the result of the filling of subshells.

  The values of the magic numbers {2, 6, 14, 28, 50, 82, 126} can be explained by a simple algorithm based upon two quantum numbers and a spin number.

Chains of Alpha Modules

The implication of the above propositions is that if one follows the spin pair linkages of the nucleons there will be chains that contain modules of the nature -n-p-p-n- or, equivalently, -p-n-n-p-. These chains may link-up thus forming a ring. Such a ring would constitute a filled nucleon shell. Here is a schematic depiction of such an alpha module ring involving four alpha modules.

The magic numbers of neutrons and protons correspond to {1, 3, 7, 14, 25, 41, 63} alpha modules. Thus a chain involving only one alpha module is just an alpha particle.

Although this has to be established, the separation distance in a spin pair of like nucleons is maintained by a repulsion between like nucleons through the strong force and a potential well for the formation of the spin pair. The separation distance for the nucleons in a neutron-proton spin pair must be maintained by rotation. So an alpha module ring rotates like a vortex ring. In the following a torus represents the surface over which nucleons move.

An alpha module vortex ring may also rotate about its axis and flip over and over like a flipped coin. The result of this motion is that a shell of a nucleus appears to be a sphere or something closely approximating one. The flipping may occur with respect to two perpendicular axes, as shown below.

In this animation the two forms of flipping are shown sequentially, but they would occur simultaneously in nature.

Upon sufficiently rapid rotation the nuclear shell would appear to look like a sphere, as shown below.

This is just as a rapidly rotating fan or propeller appears as a blurred disk. It is not feasible to show a nucleon moving along a path of a toroidal helix. That motion would effectively smear the position of a nucleon throughout a spherical shell.

Centrifugal force could distort the loop of alpha modules into an ellipse and its flipping rotation with respect to one axis would create the appearance of an oblate spheroid (pumpkin-shaped), such as the one shown below.

The rotation of an oblate spheroid about an axis perpendicular to its axis of symmetry would create the appearance of a prolate spheroid (watermelon-shaped).

The Binding Energy of Alpha Modules

An alpha particle has an extraordinary high level of binding energy compared to nuclides of lesser size. The binding energies of larger nuclides appears to be due to the formation of alpha particles within the nucleus. Generally this is not the case, although the first shell is an alpha particle. This would account for the emergence of an alpha particle in radioactive decay. But the alpha module appears to have the same binding energy characteristics as an alpha particle.

Conclusions

The formation of exclusive spin pairs of nucleons leads inexorably to the formation of chains of alpha modules. The formation of alpha module rings and their rotation and flipping leads to their nuclei appearing to be spheres or ellipsoids.

(To be continued.)

Dedicated to Warren
A friend for fiftysome years
and an always willing reader