San José State University

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Thayer Watkins
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The Empirical Performance of the Alpha
Module Model of Nuclear Structure

The Model

Nuclei are composed of protons and neutrons. The mass of a nucleus is less than the mass of its constituent nucleons. This mass deficit when expressed in energy units is called the binding energy of the nucleus. Binding energy behaves much like a loss of potential energy. When two nucleons form a spin pair the binding energy increases.

Nucleons are subject to two types of forces: those associated with the formation of spin pairs and that which is usually called the nuclear strong force. Spin pair formation is exclusive; i.e., on neutron can form a spin pair with one other neutron and with one proton and likewise for a proton. The binding energy of spin pair formations is on the order of 2 to 3 million electron volts (MeV). The so-called nuclear strong force is not exclusive but it is not all that strong when compared with the force associated with spin pair formation, being on the order of a fraction of a MeV for a single interaction. In nuclides involving a small number of nucleons the binding energy associated with spin pair formation is dominant, but in nuclides involving larger numbers of nucleons the large number of interactions makes the lnuclear strong force, even at a fraction of a MeV for each interaction, dominant.

Binding energies have been measured for almost three thousand nuclides. In order to avoid the analysis being complicated by the effect of the formations of proton-proton and neutron-neutron spin pairs only the nuclides with even number of protons and even number of neutrons are considered. There are 738 such nuclides. Even among these there is the formation of neutron-proton spin pairs to consider. When a neutron-neutron spin pair combines with a proton-proton spin pair the result is an alpha particle. An alpha particle has extraordinarily large binding energy compared with lesser structures such as a deuteron, a single neutron-proton spin pair.

Nucleon spin pairs can form more complicated structures than alpha particles. One neutron can form a spin pair with a proton which in turn can form a spin pair with another proton and that proton can form a spin pair with a neutron. Thus chains of nucleons can form which are made up of modules of the form -n-p-p-n-, or equivalently, -p-n-n-p-. These units will be called alpha modules. The number of alpha modules in a nuclide will be denoted as α. There may be excess proton spin pairs, the number of whiich will be denoted as xp. Likewise xn is the number of excess neutron spin pairs.

The binding energy of a nuclide may also be affected by the interaction of the alpha modules with each other. The number of such interactions is ½α(α−1). Likewise there may be strong force interactions among the excess protons and the excess neutrons. The numbers of these interactions are denoted by Iα, Ixp and Ixn. There is also the strong force interactions between the alpha modules and the excess protons and excess neutrons, denoted by Iαxp and Iαxn, respectively.

The binding energies for the 738 even-even nuclides was regressed on the variables cited above.

The Results

Regression Results
symb coeff t-ratio
C0 -52.08675941 -67.9
α 42.93625072 501.8
xp 18.18089664 13.8
xn 13.00631881 93.5
-0.650475821 -112.6
Ixp -4.136221139 -3.7
Ixn -0.826708798 -54.1
Iαxp -1.400873093 -13.3
Iαxn 0.394574963 46.9

The statistical performance of the model is quite good. The coefficient of determination (R²) is equal to 0.99991273 and the standard error ot the estimate (σ) is 4.76 MeV. The average binding energy for the set of observations used in the regression is 1074.17 MeV so the coefficient of variation is 0.0044, less than one half of 1 percent. The t-ratios, the ratios of the regression coefficients to their standard deviations, are quite large.

The signs of the coefficients are meaningful. Negative signs for the interactions among the alpha modules, excess protons and excess neutrons indicate the force between like particles is repulsion. Other work indicates that the nucleonic strong force charge of a neutron is −2/3 as compared to such a charge for a proton of +1. This means the nucleonic strong force charge of an alpha module is +2/3. Thus the interaction of an alpha module with an excess proton pair should be a repulsion and the negative sign for the coefficient of such interactions confirms the repulsion. On the other hand the interaction of an alpha module with a neutron pair should be an attraction and the positive value of the coefficient for such interactions confirms this.

If the electrostatic repulsion between protons is ignored and the value of −2/3 used for the nucleonic charge of a single neuton then the coefficients for the various interactions should be proportional to following values:

symb coeff
-0.44
Ixp -4.00
Ixn -1.78
Iαxp -1.33
Iαxn +0.44

By coincidence the coefficient of proportionality is roughly 1.0. Although not perfect the values are roughly in the proportion of the empirical estimates of the various coefficients.

By making the coefficient of Ia, the interactions of the alpha modules, a linear function of the number of alpha modules the coefficient of determination can be raised to 0.99993 and the standard error of estimate reduced to 4.27 MeV. This makes the coefficient of variation slightly less than 0.4 of 1 %. The t-ratio for the added variable is −13.4. Making that coefficient a function of the number of alpha modules can be justified as making the coefficient a function of the size of the nuclide and using the number of alpha modules as a proxy for nuclide size.

Regression Results
symbol coeff t-ratio
C0 -42.27112369 -42.0
a 41.52183878 317.5
xp 13.46885123 10.9
xn 12.41236785 93.7
Ia -0.522632399 -48.0
Ixp -2.733182474 -2.7
Ixn -0.852333121 -61.6
Iaxp -1.071882373 -11.0
Iaxn 0.424042286 53.9
aIa -0.001831113 -13.4

Conclusion

The model performs very well statistically and the signs of the coefficients confirm the qualitative aspect of the model, the most important of which is that through the nuclear strong force like nucleons repel each other and unlike nucleons attract.


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