﻿ The Nature of the So-Called Nuclear Strong Force
San José State University

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Thayer Watkins
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The Nature of the So-Called
Nuclear Strong Force

## The Conflation of Disparate Phenomena

• There are two types of forces involved:
• The forces associated with the formation of spin pairs of the three types, neutron-neutron, proton-proton and neutron-proton. Spin paring is exclusive. A neutron can form a spin pair with one and only one other neutron and with only one proton. Likewise a proton can form a spin with one and only one other proton and with only one neutron.
• A force due to the interaction of nucleons. It is non-exclusive and distance-dependent. It drops off faster than inverse distance-squared. An appropriate name would be nucleonic force, the force between nucleons.

The nucleonic force is not exclusive but in the interaction between two nucleons the energy associated with the formation of a spin pair is many times larger than that involved in their interaction through the nucleonic force. In a nucleus having many nucleons the magnitude of the energies of the many interactions may exceed those of the spin pair formations.

• The electrostatic (Coulomb) repulsion between protons is also involved in the structure of nuclei. It is proportional to inverse distance squared. It only affects interactions between protons. Neutrons have no net electrostatic charge but have a radial distribution of electrostatic involving an internal positive charge and a negative outer charge.
• The mass of a nucleus made up of many neutrons and protons is less than the masses of its constituent nucleons. This mass deficit when expressed in energy units through the Einstein formula E=mc² is called the binding energy of the nucleus. This binding energy has the same characteristics as the loss in potential energy involved in the formation of a nucleus. The incremental binding energies of nuclides reveal important information about the structure of nuclei. If n and p are the numbers of neutrons and protons, respectively, and BE(n, p) is the binding energy then the incremental binding energies with respect to the number of neutrons and the numbers of protons are given by:

#### ΔNBE(n, p) = BE(n, p) − BE(n-1, p) and ΔPBE(n, p) = BE(n, p) − BE(n, p-1)

• Here are some of the characteristics of nuclei revealed by incremental binding energies:
• The effects of neutron-neutron spin pair formation on binding energy  The sawtooth pattern is a result of the enhancement of incremental binding energy due to the formation of neutron-neutron spin pairs. The regularity of the sawtooth pattern demonstrates that one and only one neutron-neutron spin pair is formed when a neutron is added to a nuclide.

• The same effects occur for proton-proton spin pair formation on binding energy  • The effect of neutron-proton spin pairs is revealed by a sharp drop in incremental binding energy after the point where the numbers of neutrons and protons are equal.

Here is the graph for the case of the isotopes of Krypton (proton number 36). There is a sharp drop when the number of neutrons exceeds the proton number of 36. This illustrates that when a neutron is added there is a neutron-proton spin pair formed as long as there is an unpaired proton available and none after that. This illustrates the exclusivity of neutron-proton spin pair formation. It also shows that a neutron-proton spin pair is formed at the same time that a neutron-neutron spin pair is formed.

The case of an odd proton number is of interest. Here is the graph for the isotopes of Rubidium (proton number 37). The addition of the 38th neutron brings the effect of a neutron-neutron pair but a neutron-proton pair is not formed, as was the case up to and including the 37th neutron. The effects almost but not exactly cancel each out. It is notable that the binding energies involved in the two types of nucleonic pairs are almost exactly the same.

This same pattern is seen in the case for the isotopes of Bromine. ## The Interactions of Nucleons through the Nucleonic Force

• The most important result of the analysis of incremental binding energy is that like nucleons repel each other and unlike attract. It is found that the increments in the incremental binding energies are related to the interaction of nucleons. If the incremental binding energy of neutrons increases as the number of protons in the nuclide increases then it is evidence that a neutron and a proton are attracted to each other. If the incremental binding energy of neutrons decreases as the number of neutrons in the nuclide increases then it is evidence that the interaction of a neutron and another neutron is due to repulsion. That is to say, neutrons. The above two graphs are just illustrations but exhaustive displays are available at neutrons and protons.

• The character of the interaction of two nucleons can be represented by their possessing a nucleonic charge. If the nucleonic charges of two particles are q1 and q2 then their interaction is proportional to the product q1q2. Thus if the charges are of the same sign then they repel each other. If their charges are of opposite sign then they are attracted to each other.

Let the nucleonic charge of a proton be designated as +1 and that of a neutron as q. Thus q is the nucleonic charge of the neutron relative to that of a proton. The best estimate of q is as −2/3. In any case it is of opposite sign from that of a proton and smaller in magnitude.

• Whenever possible nucleons form spin pairs. Having established this principle it then follows that nucleons in nuclei form chains of nucleons linked together by spin pairing. Let N stand for a neutron and P for a proton. These chains involve sequences of sort -N-P-P-N- or equivalently -P-N-N-P-. Thhe simplest chain of this sort is the alpha particle in which the two ends link together. These sequences of two neutrons and two protons can be called alpha modules. They combine to form rings. A schematic of such a ring is shown below with the red dots representing protons and the black ones neutrons. The lines between the dots represent spin pair bonds. It is to be emphasized that the above depiction is only a schematic. The actual spatial arrangement is quite different. For illustration consider the corresponding schematic for an alpha particle and its spatial arrangement. • Since nucleons in nuclei form spin pairs whenever possible it is expeditious to work with the numbers of neutron-neutorn spin pairs and proton-proton spin pairs instead of the numbers neutrons and protons per se. The graph below demonstrates the existence of nucleon shells. The sharp drop off in the incremental binding energy of neutrons after 41 indicates that a shell was filled and the 42nd neutron pair had to go into a higher shell.

Maria Goeppert Mayer and Hans Jensen established a set of numbers of nucleons corresponding to filled shells of (2, 8, 20, 28, 50, 82, 126). Those were based on the relative numbers of stable isotopes. The physicist, Eugene Wigner, dubbed them magic numbers and the name stuck..

In the above graph the sharp drop off in incremental binding energy after 41 neutron pairs corresponds to 82 neutronsk, a magic number

Analysis in terms of Incremental binding energies reveal that 6 and 14 are also magic numbers. If 8 and 20 are considered the values for filled subshells then a simple algorithm expains the sequence (2, 6, 14, 28, 50, 82, 126).

• The nucleon shells are filled with rings of alpha modules. The lowest level ring is just an alpha particle. That is to say, at the center of every nucleus having two or more neutrons and two or more protons there is an alpha particle. Some nuclei are unstable and emit one and only one alpha particle.

These alpha module rings rotate in four modes. They rotate as a vortex ring to keep the neutrons and protons (which are attracted to each other) separate. The vortex ring also rotates like a wheel about an axis through its center and perpendicular to its plane. Furthermore the vortex ring also rotates like a flipped coin about two different diameters perpendicular to each other. The above animation shows the different modes of rotation occurring sequentially but physically they occur simultaneously. (The pattern on the torus ring is just the wheel-like rotation to be observed.)

Aage Bohr and Dan Mottleson found that the angular momentum of a nucleus (moment of inertia times the rate of rotation) is quantized to h(I(I+1))½, where h is Planck's constant divided by 2π and I is a positive integer. Using this result the rates of rotation were found to be billions of times per second. Because of the complexity of the four modes of rotation each nucleon is effectively smeared throughout a spherical shell. So, although the static structure of a nuclear shell is that of a ring, its dynamic appearance is that of a spherical shell. ## Nuclear Stability

• An alpha module has a nucleonic charge of +2/3=(1+1-2/3-2/3). Therefore two spherical shells composed of alpha modules would be repelled from each other if the spherical shells are separated from each other. This would be a source of instability. But if the spherical shells are concentric the repulsion can be a source of stability. Here is how that works.

Without loss of generality the force between two nucleons can be represented as

#### F = Hq1q2f(s)/s²

where s is the separation distance between them, H is a constant, q1 and q2 are the nucleonic charges and f(s) is a function of distance. For the nucleonic force it is presumed that f(s) is a positive but declining function of distance. This means that the nucleonic force drops off more rapidly than the electrostatic force between protons.

When one spherical shell is located interior to another of the same charge the equilibrium is where the centers of the two shells coincide. If there is a deviation from this arrangement the increased repulsion from the areas of spheres which are closer together is greater than the decrease in repulsion from the areas which are farther apart. This only occurs for the case in which f(s) is a declining function. If f(s) is constant there is no net force when one sphere is entirely enclosed within the other. For more on this surprising yet obvious result see Repelling spheres.

• There are nuclei in which there are neutron spin pairs in excess of the number of proton spin pairs. These are in orbits outside of the concentric spheres but held in orbit by the attraction through the nucleonic force between neutrons and the alpha modules. There are a few nuclides with excess proton spin pairs and their structure can also be explained.

## Conclusions

What conventional theory calls the nuclear strong force is made up of two disparate phenomena: Exclusive spin pairing and non-exclusive interaction of nucleons.

In a nucleus wherever possible the nucleons are linked together through spin pair formation into rings of alpha modules which rotate in four different modes at rapid rates. This rapid rotation results in each nucleon being effectively smeared uniformly throughout a spherical shell.

The nucleons are organized in spherical shells containing at most certain numbers of nucleons. These nuclear magic numbers are explained by a simple algorithm.

Dynamically a nucleus is basically composed of concentric spherical shells which repel each other. This mutual repulsion results in a stable arrangement in which the centers of the concentric spherical shells coincide.

Thus a nucleus is held together by the linkages created by the formation of spin pairs. The rings of alpha modules rotate to create the dynamic appearance of concentric spherical shells which are held together through the repulsion of the nucleonic forces. Neutron spin pairs outside of the concentric spheres are held by their attraction to the concentric spheres. So all of the nuclear forces are involved in holding a nucleus together

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