San José State University

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Thayer Watkins
Silicon Valley
& Tornado Alley
U.S.A.

An Empirical Testing of the Alpha
Module Model of Nuclear Structure

Introduction

The nature of what is meant by alpha module will be explained below. The name alpha module model is an incomplete description of the model being considered. More generally the model explains the binding energies of nuclides as being made up of two parts; one part is due to the formation of substructures within nuclei and the other part is due to the nuclear strong force interactions of the nucleons within nuclei. The substructures within nuclei are formed as a result of spin pairing. A neutron whenever possible forms a spin pair with another neutron. It also, whenever possible, forms a spin pair with a proton. Likewise a proton, whenever possible, forms a spin pair with another proton and with a neutron.

If one considers the linkages formed by spin pairing one sees that chains are formed. For example, a neutron links to a proton which links to another proton and that proton links to a neutron. So such chains are made up of modules of the form -n-p-p-n-, or equivalently, =p=n-n-p-. The chains have to close into rings of alpha modules. Here is symbolic depiction of a ring of alpha modules.

The smallest such chain would be simply an alpha particle. Thus in a nucleus having enough nucleons to create an alpha module there would be an alpha particle, but only one. The shell theory says that there are filled shells of neutrons and protons. The capacities of these filled shells are conventionally taken to be {2, 8, 20, 28, 50, 82, 126}. The separate shell capacities would then be {2, 6, 12, 8, 22, 32, 44}. Elsewhere it is argued that the capacities are {2, 4, 8, 14, 22, 32, 44} with the filled shell totals being {2, 6, 14, 28, 50, 82, 126} with 8 sand 20 being in the nature of filled subshells. The shell capacities correspond to alpha modules of {1, 2, 4, 11, 16, 22}.

Mass Deficits, Binding Energies
and the Mass of the Neutron

The masses of charged structures can be measured through their trajectories in a magnetic field. It is found that the mass of a nucleus is less than the mass of the protons and neutrons which make it up. This is called the mass deficit of a nucleus and when it is expressed in energy units it is called the binding energy of the nucleus. All of this depends upon the mass of the neutron, the one particle whose mass cannot be measured directly.

The mass of the neutron is deduced from a property of a deuteron (a nucleus made up of one proton and one neutron). When a deuteron is formed a gamma ray of energy 2.2 million electron volts (MeV) is emitted. The conventional estimate of the mass of a neutron is based upon the assumption that the binding energy of a deuteron is the same as the energy of the gamma ray emitted upon its formation. However in transitions of electrons the energy of the photon emitted is only half of the change in energy of the electron. The half ratio applies only when the force on a particle is strictly inversely proportional to the square of distance. The forces in nuclei may follow a different formula, but there would be some additional binding energy beyond that of the emitted photons. Thus the conventional estimate of the mass of the neutron is an underestimate. Therefore the conventional binding energies are also underestimates. This evidenced by the binding energy of one nuclide, Beryllium 6, being a negative figure.

The Testing of the Model

For the 2929 nuclides the following variables were computed which represent the formation of substructures.

To represent the interactions between nucleons the following variables were computed.

The effect of the formation of a substructure on binding energy is not a constant amount applying under all conditions. For x formations it is in the nature of (a − b/x) where a and b are coefficients. When this is multiplied by x the result is (ax − b). This means that the regression equation for binding energy should include a constant that represents the cumulative total of the b coefficients for all of the substructures.

Here are the regression equation coefficients and their t-ratios (the ratios of the coefficients to their standard deviations).

The Results of Regression Analysis
VariableCoefficient
(MeV)
t-Ratio
Number of
Alpha Modules
42.64120923.0
Number of
Proton-Proton
Spin Pairs
13.8423452.0
Number of
Neutron-Neutron
Spin Pairs
12.77668165.5
Number of
Neutron-Proton
Spin Pairs
13.6987565.3
Proton-Proton
Interactions
−0.58936−113.8
Neutron-Proton
Interactions
0.3183195.8
Neutron-Neutron
Interactions
−0.21367−96.6
Constant49.37556−112.7
0.99988

It should be noted that the nuclides of a single proton and a single neutron were left out of the analysis. Also that there is a great difference among the frequencies of the extra spin pairs. There are 2919 with an alpha module and only 10 without. There are 2668 nuclides with extra neutron-neutron spin pairs, but only 164 out of the 2929 nuclides which have one or more extra proton-proton spin pairs. There are 1466 with an extra neutron-proton spin pair.

Results and Conclusions

The coefficient of determination (R²) for this equation is 0.9998825 and the standard error of the estimate is 5.47 MeV. The average binding energy for the nuclides included in the analysis is 1072.6 MeV so the coefficient of variation for the regression equation is 5.47/1072.6=0.0051. Most impressive are the t-ratios. A t-ratio of about 2 is considered statistically significant at the 95 percent level of confidence. The level of confidence for a t-ratio of 923 is beyond imagining.

It is notable that the coefficients for all three of the spin pair formations are roughly equal. They are all larger from what one would expect from the binding energies of small nuclides.

The regression coefficients for the strong force interactions have some especially interesting implications. The force between to particles with nuclear strong force charges of Q1 and Q2 is of the form

F = HQ1Q2f(s)/s²

where H is a constant, s is the separation distance and f(s) is a declining function of s, probably exp(−s/s0).

Let the strong force charge of a proton be taken as 1 and that of a neutron as q, where q might be a negative number. The nuclear strong force interactions between neutrons is proportional to q², those between neutrons and protons would be proportional to q and those between protons would be proportional to 1. Thus the ratio of the coefficient for neutron-neutron interactions to that for neutron-proton interaction would be equal to q. The value of that ratio is −0.67127. This would be confirmation of the value of −2/3 found in previous studies. Thus the nuclear strong force between like nucleons is repulsion and attraction between unlike nucleons.

The values involving proton-proton interactions are most likely affected by the influence of the electrostatic repulsion between protons. That force would be

F = Hf(s)/s² + Ke²/s²

where e is the electrostatic charge of a proton and K is the constant for electrostatic force. The above formula can be rearranged to

F = [Hf(s)/s²][1 + Ke²/(Hf(s))]
and treated as
though it is
F = HQ²f(s)/s²

where the strong force charge of a proton is effectively Q rather than 1. Thus the effective strong force charge of the proton taking into account the electrostatic repulsion can be represented as (1+d²)½ where d² represents the average value of Ke²/(Hf(s)) at the separation distances involved in nuclei. With the electrostatic repulsion included, the ratio of the coefficient for neutron-neutron interaction to that of proton-proton interaction should be q²/(1+d²). The value of that ratio 0.36255. This means that q/(1+d²)½ is equal to √0.36255=±0.60213. The ratio of the coefficient for neutron-neutron interactions to that of neutron-proton interaction is also equal to q/(1+d²)½ and that value, as previously noted, is −0.67127.

The ratio of the coefficient for neutron-proton interactions to that for proton-proton interactions should also be q/(1+d²)½ and that ratio is −0.54010. So there are three estimates of the value of q/(1+d²)½, with the average being −0.60450. If this average is used along with a value of −2/3 for q then

(1+d²)½ = (−2/3)/(−0.60450) = 1.10284

This means that the electrostatic force between protons is about one tenth of the nuclear strong force between them.

All in all the test is a resounding confirmation of the Alpha Module Model of nuclear structure.

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