San José State University

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Thayer Watkins
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 A Model of a Quasi-Alpha Particle Structure of Nuclei

One of the original theories of the structure of nuclei is that whenever possible the nucleons are combined into alpha particles and then the alpha particles are combined to form the nucleus. There is much evidence in favor of this structural theory. Most of the binding energies of nuclei can explained as being due to the formation of alpha particles. When some nuclei decay there are alpha particles ejected. There are however problems. Alpha particles having an electrostatic charge of +2 should repel each other. They also have a positive nucleonic (strong force) and should repel each other. A small separation distances the neutrons of one particle may be attracted to the protons of another, but that possibility is in conflict with the rotation required to keep neutrons and protons apart.

## Spin Pairs

Neutrons form spin pairs and so do protons, but they also can form neutron-proton spin pairs. There is an increase in binding energy associated with the formation of spin pairs. This increase in binding energy is due to a loss of potential energy when the spin pair is formed. The potential energy for the pair formation is a so-called square well potential; i.e., it is a constant negative value up to a limited separation distance and zero thereafter.

#### V(s) = −K for s≤slimV(s) = 0 for s>slim

The potential associated with the strong force is a function of distance. For a repelling force it is an inverse function of distance. Neutrons repel each other. The potential energy function for two neutrons would look like the graph shown below.

The units are arbitrary. The system would behave dynamically like a ball set up on the curve. If started at a separation distance below slim it would roll to the notch at slim. Thus the separation distance of the neutrons in a neutron-neutron spin pair would simply be slim.

Two protons forming a spin pair would behave similarly. The separation distance would be the slim for a proton-proton spin pair. For a neutron-proton spin pair it is a different story.

A ball set upon this potential energy curve would roll down to a separation distance of zero. To maintain a separation of the neutron and proton it is necessary to have the particles rotate about their center of mass.

As stated above neutrons form spin pairs and so do protons, but they also can form neutron-proton spin pairs. This means that there is double linkage for each nucleon. This allows for the creation of closed chains involving modules of two neutrons and two protons. Such a module, called a quasi-alpha particle, is illustrated below with the black spheres representing neutrons and the red ones protons. The lines denote the linkages involving spin pairs. The system would rotate about an axis midway between the neutrons and the protons.

The case of four quasi-alpha particles in a ring is shown below.

The rotation is about the ring that represents the circle on which the centers of masses of the neutron-proton pairs lie; i.e. the neutron pairs and the proton pairs are alternately inside and outside of the ring.

Such closed chains would constitute linked neutron and proton shells. The successive capacities of the shells are 2, 4, 8, 14, 22, 32 and 44 when the magic numbers are taken to be 2, 6, 14, 28, 50, 82 and 126. The capacities in terms of nucleon pairs are 1, 2, 4, 7, 11, 16 and 22. There is a quantum mechanical justification for this series of numbers. Note that if 1 is subtracted from each the result is 0, 1, 3, 6, 10, 15, 21. This series is just the cumulative sums of the series 0, 1, 2, 3, 4, 5, 6.

## The Three Dimensional Nature of the Chains of Quasi-Alpha Particle Modules

The closed chains of nucleons do not necessarily lie in a flat band as depicted above. The corresponding depiction of an alpha particle would be the image on the left below but the better depiction of an alpha particle is such that the nucleons are located at the vertices of a tetrahedron, as shown on the right below.

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A more elegant version of the structure of an alpha particle is shown below.

The second shells for neutrons and protons would have the following structure.

The linkages in the above arrangement are shown below.

Such an arrangement extended to the third shells previously depicted would look like this.

In this diagram the red circles each represent two protons, one behind the other. The protons are linked as spin pairs. One link going to a red circle represents a linkage to one proton and a link going out represents a linkage from the other proton to a neutron.

## The First Shell

The model indicates that the first shells of neutrons and protons would together be a single alpha particle. When nuclei decay by alpha particle emission it is a single alpha particle. Thus the center of any nucleus would be an alpha particle. If all of the nucleons, wherever possible, were organized into alpha particles there would likely be some cases of the emission of multiple alpha particles or small nuclei consisting entirely of alpha particles. These are not the case.

## Conclusions

The arrangements of quasi-alpha module can account for the binding energies and the linkages within the nucleus. The binding energy of a quasi-alpha particle would be essentially the same as that of an alpha particle. Thus such arrangements can account for the binding energies and the linkages within the nucleus.

(To be continued.)