San José State University
Thayer Watkins
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What Determines the Magnitude
of the Binding Energy Due to
Nucleonic Spin Pairing

Here is the situation. A nucleon, say a neutron, has a nucleonic charge and spins at a fantastically high rate. That spinning generates a field just as a spinning electrical charge generates a magnetic field. The magnetism-like field gives the neutron a north pole and a south pole.

One neutron will link with another neutron with north poles matching up with south poles. But the neutrons are repelled from each other due to their like nucleonic charge. The neutron will also spin pair with a proton, but there is an attraction involved instead of a repulsion. Therefore stability requires a rotation of the neutron and proton about their center of mass.

Chains form involving modules of the form -P-N-N-P-, or equivalently -N-P-P-N-. These chains close to form rings which rotate like wheels and flip like coins in addition to being vortex rings. Such rotations give a ring the dynamic appearance of a spherical shelll. A nucleus is made up of a set of such concentric shells, with the innermost one being an 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 rotates like a wheel about an axis through its center and perpendicular to its plane. 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 to allow 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 are 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 structure is that of a spherical shell.

At rates of rotation of billions of times per second all that can ever be observed concerning the structure of nuclei is their dynamic appearances. This accounts for all the empirical evidence concerning the shape of nuclei being spherical or near-spherical.

Taking into account the shell structure of nuclei this model explains 99.995 percent of the variation in the binding energy of the almost three thousand known nuclides. However that version of the model involves a large number of independent variables which makes displaying the regression results awkward. Instead the simplest version of the model will be utilized.

The regression results for that version are:

Regression Results for Binding Energy
# of PP10.44441.4
# of NN12.682147.7
# of NP9.69068.8

All of the coefficients for the numbers of spin pairs are positive indicating that spin pairing is an attractive that adds to binding energy. The coefficients for the interaction of protons with protons and neutrons with neutrons are negative indicating these interactions are repulsions. The coefficient for the interaction of unlike nucleons (neutrons with protons) is positive indicating that such interactions involve an attraction. This is the common rule that like charges repel and unlike charges attract each other.

A t-ratio for a regression coefficient is the ratio of its value to its standard deviation. The t-ratio of a regression coefficient must be of magnitude 2 or greater for it to be considered significantly different from zero at the 95 percent level of cf confidence.

The coefficient of determination (R²) for the regression equation is 0.99985, a quite respectable achievement.

The coefficients for the interaction contain some important inormation. If the nucleonic charge of a proton is taken to be 1 and that of a neutron is denoted as q, then the interaction of protons and neutrons should be proportional to 1·q whereas the interaction of neutrons with neutrons should be proportional to q². Thus the ratio of coefficient for neutron interactions to the coefficient for proton-neutron interaction should be equal to q. The value of this ratio is −0.6691, amazingly close to −2/3.

It is a bit perplexing to have such a large constant term for the regression. It was rerun with constant term eliminated.

Regression Results for Binding Energy
# of PP4.3108.3
# of NN13.83577.0
# of NP10.31234.7

The results are similar. The estimate of q, the nucleonic charge of the neutron relative to that of the proton is −0.693.

According to the above regression equations the interaction of unlike nucleons increases binding energy but the interaction of like nucleons decreases it. The interaction coefficients may be used to compute the net effect of the nucleonic interaction on bindinding energy. This net effect was divided by binding energy to get the relative influence of nucleonic interaction on binding energy. The frequency distribution of this quantity is shown below.

As can be seen predominantly the effect of the nucleonic interactions is to decrease binding energy. The average of this relative net effect of nucleonic interactions is −0.159. This means that overwhelmingly what holds a nucleus together is the spin pairing of its nucleons.

The Strength of the Magnetism-like
Field Generated by the Rotation
of the Nucleonic Charges

Special Relativity requires the movement of an electrical charge generate a magnetic field. That same argument requires the existence of a magnetism-like field for any two-valued charge.

David Griffiths in his book, Introduction to Electrodynamics provides a magnificant derivation of the magnetic field due to a spinning spherical shell of electric charge. This derivation applies equally well to a spinning shell of nucleonic charge.

Griffiths' formula for the vector potential A(r, θ, φ) of a spherical shell of radius R carrying a uniform surface charge of σ and spinning at an angular rate of ω is

⅓μ0Rωσr·sin(θ)φ^ for r≤R
⅓μ0R4ωσ(sin(θ)/r²)φ^ for r≥R

where φ^ stands for the unit vector in the positive φ direction.

The magnetic field B is given the curl of the vector potential; i.e.,

B = ∇×A

Within the spherical shell B is uniform and points in the direction of the spin vector (which coincides with the z axis).

The functional form would be the same for the magnetism-like field associated with a spinning nucleonic charge. The spin rates would be the same for the electrostatic and nucleonic charges. The magnitude of the associated vectors can only differ the basis of the spatial charge densities and the parameter corresponding to μ0, the magnetic permeability of free space. The units in which charge density is measured affects the magnitude of the permeability parameter. Thus it is only the product μ0σ which is relevant.

(To be continued.)

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