﻿ The Spin Pairing of Nucleons Due to the Magnetic Fields Generated by their Spinnings
San José State University

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Thayer Watkins
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The Spin Pairing of Nucleons Due to the
Magnetic Fields Generated by their Spinnings

A major factor in what holds a nucleus together is the formation of nucleonic spin pairs; neutron-neutron, neutron-proton, and proton-proton. These spin pairs form whenever possible but they are exclusive in the sense that a neutron may pair with one other neutron and with one proton but no more. The same holds for a proton.

The binding energies involved in the formation of the three types of nucleonic spin pairs are approximately equal and an order of magnitude larger than that due to the nonexclusive interactions between nucleons.

The first question at issue is what is the nature of the spin pairs.

One possible candidate for what holds spin pairs together is the magnetic fields of the nucleons. Moving electrostatic charges generate magnetic fields. Rotating charges are especially relevant for generating magnetic fields,

Consider a space described by the spherical coordinates (r, φ, θ). A spherical charge of radius R centered at the origin with surface charge density σ spinning about the axis φ=0 at a rate of Ω times per second generates a magnetic field according to the formula

#### For r≥R B = μ0RΩσ/(6π)(R/r)³[2cos(φ)ur + sin(φ)uφ]

where ur and uφ are the unit vectors in the directions of increasing r and φ, respectively.

The field is perfectly symmetric with respect to the angle θ

For the interior of the sphere B is constant and points to the spin axis. Its value is

#### For r≤R B = 2μ0RΩσ/(6π)[cos(φ)ur − sin(φ)uφ]

There is a discontinuity in B at r=R.

## The Attraction of Spinning Spherical Charges

The force between two magnets which have magnetic field strengths of B1 and B2 at their poles is

#### F = CB1B2R4[1/s² + 1/(2R+s)² − 1/(R+s)²

where C is a constant, R is the spatial scale of the magnets and s is the separation distance between the poles.

## The Alignment of Spinning Spherical Charges

The spinning spherical charges are like magnets with two poles. Since opposite poles attract two spinning spherical charges may form some sort linked entity. However the effect of the charges must be taken into account. It is alleged that if the spheres have to have opposite spins. As shown below this is not necessarily true. It depends upon what spatial arrangement involves the minimum energy. This depends upon the charges of the spheres among other things.

If they have opposite charges then they must have the same direction of spin.

There is the possibility that two charged spheres could align themselves like two bar magnets with the opposite poles matched together. would be like the depiction shown below where the yellow arrows denote the direction of the spin. If the spheres have the same charge the electrostatic repulsion in the above arrangement would be acute but the separation between the poles of the spinning spheres would relatively large.

The alternative is an end-to-end alignment as shown below. For spheres of the same charge the end-to-end, spin-pole to spin-pole arrangement would release more potential energy than any other. This is due to bringing the attracted magnetic poles together and separating the repelling charges as much as possible within the constraint of a spin pair. It will be found to be a different matter for spheres of different charges.  The different colors denote different charges. different charges.

The fact that one nucleon can form at most two spin pairs suggests that the end-to-end alignment might be the one that prevails, However other considerations suggest the arrangement involving end-to-end for like charges and side-by-side for unlike charges. Those other considerations have to do with what would occur when the odd charged sphere links to another sphere of its own charge.

## The Distributions of Electrostatic Charges of Nucleons

Although a neutron has zero net electrostatic charge that does not mean it has no charges at all. The graph gives the distributions. A proton has a measured magnetic moment of 2.793 magnetrons and a neutron's is −1.913 magnetrons. Thus although a neutron has no net electrostatic charge the fact that its negative charge is located at a greater distance from its spin axis than is its positive charge results in its having a magnetic field of the same sort that a spinning sphere of negative charge would have. Therefore previous description concerning spinning spheres of opposite electrostatic charges is relevant for protons and neutrons. But there is another consideration.

## Nucleonic Charge

An investigation of the binding energies of nuclides indicates that there is a force associated with the interaction of nucleons. In contrast to the spin pairing the interaction force is nonexclusive. The evidence is that the interaction of unlike nucleons is an attraction but the interaction of like nucleons is a repulsion. Its magnitudes can be explained by protons and neutrons having a charge, which can appropriately be called a nucleonic charge. If the nucleonic charge is taken to be +1 then the nucleonic charge of a neutron is −2/3>.

What conventionally is called the nuclear strong force is a conflation of disparate phenomena; i.e., the attractive exclusive force of spin pair with interaction force in which like nucleons are repelled from each other and unlike ones are attracted.

The previous material involving electrostatic charges more appropriately applies to spheres of nucleonic charge. It does also apply to spheres of electrostatic charge but is limited due to the fact that neutrons have zero net electrostatic charge.

The application of the above material presumes that there is a field generated by rotating spheres of nucleonic charge which is analogous to the magnetic field generated by rotating spheres of electrostatic charge. This is plausible but is not supported by any empirical evidence so far other than the phenomenon of nuclear spin pairing.

However there is an argument that special relativity requires the existence of magnetism to accompany the electric field. If that argument proves to be valid then it is quite likely that it would hold for any charged field, including the one that accounts for the interaction of nucleons. This is because special relativity concerns the properties of spacetime itself rather than a specific force field. This idea is explored in Nuclear Magnetics. The involvement of the nucleonic force would account for the power of the spin pair formation because at the distances involved in a nucleus the nucleonic force is several times more powerful than the electrostatic force.

## Conclusion

The spin pairing of nucleons could involve linkages resulting from the magnetic fields generated by their spinning. The closest approach of two nucleons of like type appears to be pole-to-pole (end-to-end) and side-by-side for opposite type nucleons.

There may be a field like the magnetic field which is generated by the rotation of nucleonic charges which is also involved in the spin pairing of nucleons.