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Empirical Investigation of the Alpha
Module Model of Nuclear Structure

From the very beginning of the search for the spatial structure of nuclei there has been the theory that the nucleons in a nucleus are, wherever possible, organized into alpha particles. The support for this theory was primarily that alpha particles show up in nuclear reactions such as the emission of alpha particles and as a fragment of nuclear fission. It seemed unlikely that such alpha particles would appear unless they already existed within the nucleus before the reaction. Another element of support for the existence of alpha particles within nuclei is the magnitude of the binding energies.

For nuclides too small for an alpha particle to be formed the binding energies are at most 7 to 8 million electron volts (MeV), but the binding energy for an alpha particle is an extraordinary 28 MeV. The binding energies for nuclides large enough to form an Talpha particle the binding energy is roughly 28 MeV times the number of alpha particles that could be formed. But the alpha particle substructure theory of nuclei does not get enough further support to be accepted as the explanation of the structure of nuclei.

A variation on the alpha particle substructure theory has been developed that offers an explanation for more nuclear phenomena. Neutrons and protons are subject to two types of forces. One is spin pair bonding and the other is the nuclear strong force. Spin pair bonding is exclusive in the sense that a neutron can form a spin pair with one other neutron and it can also form a spin pair bond with one proton. The same applies for a proton. Thus neutrons and protons can form chains involving modules made up of two neutrons and two protons such as n-p-p-n and equivalently p-n-n-p. Such a chain is shown below.

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 and 20 being in the nature of filled subshells. The shell capacities correspond to alpha modules of {1, 2, 4, 11, 16, 22}.

The Explanation of Nuclear Binding Energies

Within a nucleus, in addition to alpha modules, there can be neutron-neutron spin pairs, proton-proton spin pairs and neutron-proton spin pairs. The formation of these substructures would contribute to the binding energy of a nuclide. There also would be interactions of the substructures that would contribute to the binding energy. The alpha modules may interact with each other. If #a is the number of alpha modules the number of interactions is ½#a(#a-1). If #nn is the number of neutron-neutron spin pairs then the number of interactions of alpha modules and neutron-neutron spin pairs is #a#nn and likewise for the number of alpha module interactions with proton-proton spin pairs. Then there would be the number of neutron-neutron spin pair interactions with each other, ½#nn(#nn-1) and so on. It should be noted that there is a great difference among frequencies of the extra spin pairs. There are 2668 nuclides with extra neutron-neutron spin pairs, but only 164 out of the 2931 nuclides which have one or more extra proton-proton spin pairs, There are 1466 with an extra neutron-proton spin pair.

According to the theory previously developed if the strong force charge of a proton is taken to be +1 then that of a neutron is −2/3. Thus the net strong force charge of an alpha module is +2/3. Alpha particles repel each other as do other alpha modules. The coefficient for the alpha module interactions should be negative. On the other hand alpha modules, with a charge of +2/3, and neutron-neutron pairs with a charge of −4/3 attrack each other and the coefficient for their interactions should be positive. Proton-proton spin pairs have a strong force charge of +2 repel alpha modules and the coefficient for their interactions should be negative. A neutron-proton spin pair has a net strong force charge of 1/3 so they repel alpha modules and the coefficient for such interactions should be negative.

Since neutrons repel each other through the strong force the coefficient for the interactions of neutron-neutron pairs with each other should be negative. The same applies for proton-proton spin pairs.

There could also be binding energy effects due to the interaction of a singleton neutron or a singleton proton with the other substructures. There are 1466 nuclides with a singleton neuton and 1460 with a singleton proton. Let #sn and #sp denote the number of singleton neutrons and protons, respectively. Only one of the two can be 1 because two of them would form a neutron-proton spin pair. The coefficient for the interaction of alpha modules with a singleton neutron should be positive and with a singleton proton negative.

The regression equation obtained using the above scheme for the binding energies of the 2931 nuclides is as follows.

BE = 39.64#a + 13.65#nn -4.64#pp -11.51#np
- 0.54(½#a(#a-1)) + 0.35#a#nn - 00.62#a#pp + 1.51#a#np
- 0.76(½#nn(#nn-1)) + 6.76(½#pp(#pp-1)) + 0.19#a#sn -1.03#a#sp

(t-ratios)
[517.0] [84.1] [-3.2] [-13.7]
[-84.9] [35.4] [-5.1] [13.7]
[-42.2] [5.4] [14.3] [-9.9]

R² = 0.999912

The magnitude of a t-ratio (ratios of a coefficient to its standard deviation) must be at least 2 in order for the coefficient to be significantly different from zero at the 95 percent level of confidence. As can be seen the t-ratios are all well above the level of 2.

The predictions for the signs of the interaction terms are borne out except in the case of the interaction of alpha modules and the neutron-proton spin pairs and the interaction of the proton-proton spin pairs. Also there should have been positive values for the formation of proton-proton spin pairs and neutron-proton spin pairs. The coefficients for the interactions of the neutron-neutron and proton-proton spin pairs with singleton neutrons and singleton protons were not statistically significantly different from zero at the 95 percent level of confidence and those variables were eliminated from the regression analysis.

Conclusions

There is support for the alpha module theory of nuclear structure but it is not complete. The predictions are based primarily upon the notion of strong force charges for the proton and neutron being +1 and −2/3, respectively. The signs of crucial regression coefficients such as the interaction of alpha modules and neutron-neutron spin pairs are as the theory of strong force charges predicts. The conventional theory that all nucleons attract each other would predict all of the regression coefficients to be positive. On balance there is more support for the alpha module theory and strong force charges than for the conventional theory.

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