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The Statistical Explanation of the Excess
Binding Energies of the Alpha Nuclides

The masses of nuclides are less than the sum of the masses of the protons and neutrons which they contain. This mass deficit translated into energy equivalence is called the binding energy of the nuclide. The binding energy of a nuclide could be composed of two components. One component could be the binding energy of substructures which are composed of protons and neutrons and another the binding energy due to the potential energy involved in putting these substructures together into an arrangement. Previous work indicates that the protons and neutrons within a nucleus form alpha particles whenever possible. Any additional protons and neutrons beyond the alpha particles substructures form pairs; neutron-neutron, neutron-proton and proton-proton. These pair formations are not mutually exclusive; i.e., a neutron may form a pair with a proton as well as with another neutron.

A major component of the binding energy of a nuclide would then be that due to the formation of alpha particles and nucleon pairs. The rest of the binding energy would be due to the configuration of those alpha particles and nucleon pairs and any excess protons or neutrons. This binding energy will be referred to as excess binding energy.

There are accepted values of binding energies for alpha particles (28.3 million electron volts (MeV) and for neutron-proton pairs (2.2 MeV) but no accepted value for neutron pairs.

The excess binding energy for those nuclides that could contain an integral number of alpha particles, hereafter called alpha nuclides, has an interesting form.

The above graph suggests that there are shell structures of the alpha particles. A shell is a collection of particles with the same quantum number(s) and hence at the same distance from the center of the nucleus. There is no significant increase in binding energy for two alpha particles but for three there is. The additional binding energy for the number of alpha particles above two appears roughly constant at about 7.3 MeV per additional alpha particle until a level of 14 alpha particles is reached. Thereafter the increase is about 2.7 MeV per additional alpha particle.

In order to investigate the numerical relationship more precisely a bent-line regression equation is fitted to the data. The bend points are based upon the so-called magic numbers, but including 6 and 14 and leaving out 8 and 20. The form of the regression equation is

XSBE = c0α + c1u(α-3) + c2u(α-7) + c3u(α-14)

where α for a nuclide is the number of alpha particles it could contain. The excess binding energy XSBE is the binding energy of the nuclide less 28.295*α. A variable u(α-k) is 0 if α≤k and (α-k) otherwise.

The result of the regression is

XSBE = 1.55345α + 6.64855u(α-3) −1.10110u(α-7) −4.37015u(α-14)
[5.6] [12.0] [-2.7] [-20.3]

The figures in square brackets, [ ], are the t-ratios for the coefficients. For a regression coefficient to be significantly different from zero at the 95 percent level of confidence its t-ratio must be greater than about 2.0 in magnitude. As seen above all of the coefficients are significantly different from zero. However when a variable u(α-10), corresponding to a nuclear magic number of 20, is included its coefficient is not statistically different form zero.

The coefficient of determination for the regression equation is 0.999169, thus indicating that 99.9161 percent of the variation in XSBE is explained by the variation in the regression equation variables. The closeness of the fit of the regression equation is revealed by a plot of the regression estimates along with the data as a function of the number of alpha particles in the nuclides.

The fit is so close that the data (the blue line) is almost everywhere covered up by the line for the regression estimates.

The incremental increase in excess binding energies for the various shells are found as the cumulative sums of the regression coefficients; i.e.,

The Incremental Increases in Excess Binding Energies
of Additional Alpha Particles in the Various Shells
Alpha
Shell
1 to 23 to 67 to 1314 to 25
Incremental
Binding
Energy
(MeV)
1.55348.202007.100892.7307

Thus what appeared to be a single shell from 3 alphas to 13 is two shells.

Conclusion

More than 99.9 percent of the variation in excess binding energy is explained by a model of nuclear shells.


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