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The Separation Distance In Spin Pairs of Like-Nucleons

San José State University

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Alley USA

The Separation Distance In
Spin Pairs of Like-Nucleons

Consider two nucleons of a like nature each with a charge q of some type and rate of rotation ω.
Because they are of a like nature the are repelled from each other and the repulsive force is given by

F = Hq²f(s)/s²

where s is the separation distance of the centers of the nucleons, H is a constant and f(s) is a monotonically
non-increasing function. This function f(s) is f(s)=1 for electrical charge and may be exp(−s/σ)
for nucleonic charge.

Special Relativity requires that associated with a moving charge of any type there is a magnetism-like
field. The spinning nucleons then generate a magnetism-like field with poles which aligns the spins and attracts the nucleons
together. The magnetic field intensity B for a spinning sphere of electrical charge is given by the formulas

B(r, φ, θ) = ∇×A(r, φ, θ)
where the vector potential A is
A = (μ_{0}qω/6π)(R/r)²sin(φ)θ^

where μ_{0} is the permittivity of space, R is the radius of the spherical charge and θ^ is the unit vector is the
direction of increasing θ. The important thing is that the magnetic field intensity due to one spinning sphere of charge is proportional
the total charge q of the sphere, say

B(r, 0, 0) = K(R/r)²qω

where K is a constant. The force F_{M} of attraction between two spinning spheres each with charge q is proportion to the product of the
field intensities.

F_{M} = L(R/r)^{4}q²ω²

The two nucleons would assume a configuration in which the repusion of their charges is balanced by the attraction due the magnetism-like
field due to their spinning; i.e.,

F = F_{M}
Hq²f(s)/s² = L(R/r)^{4}q²ω²

The distance r is equal to s/2. Both sides of the above equation are proportional to q² and this term may be cancelled out. Thus
the above equation reduces to

Hf(s)/s² = 16L(R/s)^{4}ω²
or, further to
Hf(s)s² = 16LR^{4}ω²

This equation has a solution for separation distance that is independent of charge.
For electrical charge for which f(s)=1 the solution is

s = 4(L/H)^{½}R²ω

Spin pairs of nucleons with different charges
but the same radius and spin rate would have the same separation distance. Thus spin pairs of neutrons and of protons would have the same
separation distance.