San José State University

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The Empirical Performance of the Alpha
Module Model of Nuclear Structure
Using the Dataset of All Nuclides

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., one neutron can form a spin pair with one other neutron and with one proton and no more. It is 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, the potential energy for it 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 potential energy associated with the nuclear strong force, even at a fraction of a MeV for each interaction, dominant.

Binding energies have been measured for almost three thousand nuclides. 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 (28.3 MeV) compared with lesser structures such as a deuteron, a single neutron-proton spin pair, with a binding energy of 2.225 MeV.

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 x#pp. Likewise x#nn is the excess number of neutron spin pairs. There can also be a singleton neutron or a singleton proton in a nucleus. These zero-or-one variables are denoted as #n and #p.

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 proton pairs and the excess neutron pairs. The numbers of interactions between the alpha; modules and the excess neutron pairs, excess neutron-proton pairs and excess proton pairs are denoted by αI#nn, αI#np and αI#pp, respectively. There is also the strong force interactions between the alpha modules and a singleton neutron or a singleton proton, denoted by #αIn and αIp, respectively.

The binding energies for the 2931 nuclides was regressed on the variables cited above.

The Results

Regression Results
SYMB COEFF T-RATIO
#a 48.00 322.1
#nn 29.47 3.4
#np 18.83 9.1
#pp 46.18 1.4
#aI#a -0.85 -127.3
#aI#nn 1.56 0.17
#aI#np 0.64 7.3
#aI#pp -9.53 -0.2
#aI#n 0.23 146.1
#aI#p -1.07 -18.5
#nnI#nn -24.4 -2.4
#nnI#np 1.61 0.2
#npI#pp -5.29 -0.2
#ppI#pp -49.21 -1.0
#nnI#n -1.16 -0.2
#ppI#p -2.78 -0.1
C0 -52.16 -24.4

The statistical performance of the model is quite good. The coefficient of determination (R²) is equal to 0.9977 and the standard error ot the estimate (σ) is 23.85 MeV. The average binding energy for the set of observations used in the regression is 1072 MeV so the coefficient of variation is 0.022, about 2 percent. The t-ratios, the ratios of the regression coefficients to their standard deviations, are quite large for some variables such as the number of spin pairs of various types but quite small for the interactions involving the minor constituents.

The signs of the coefficients are meaningful. Negative signs for the interactions among the alpha modules, excess neutron pairs and excess proton pairs indicate the force between like particles is repulsion. Positive signs for interactions indicate an attraction between two types of constituents.

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:

Theoretical
Relative Values
of Interaction
Coefficients
Symbol relative
coeff
#αI#α -0.44
αI#nn +0.88
αI#np -0.22
#αI#pp -1.33
αI#n +0.44
αI#p -0.67
#nnI#nn -1.78
#nnI#np +0.44
#ppI#np -0.67
#nnI#n -0.88
#ppI#pp -4.00
#ppI#p -2.00
Empirical Values
of Interaction
Coefficients
and Their t-Ratios
Symbol regress
coeff
t-ratio
#aI#a -0.85 -127.3
#aI#nn +1.56 0.1
#aI#np +0.64 7.3
#aI#pp -9.53 -0.2
#aI#n +0.23 146.1
#aI#p -1.07 -18.5
#nnI#nn -24.45 -2.4
#nnI#np +1.61 0.2
#ppI#np -5.29 -0.2
#nnI#n -1.16 -0.2
#ppI#pp -49.21 -1.0
#ppI#p -2.78 -0.1

Although not perfect the values are roughly in the proportion of the empirical estimates of the various coefficients. The signs are all in agreement except for that of #aI#np, the interaction of alpha modules with neutron-proton spin pairs, where the theory says it should be −0.22 but the empirical estimate is +0.64. However many of the regression coefficients are not statistically different from zero at the 95 percent level of confidence. For a coefficient to be significantly different from zero at the 95 percent level of confidence the t-ratio must be 2 or larger.

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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