There are nuclides with the neutron and proton shells entirely filled and having additional neutrons in an outer shell. Such neutrons are called halo neutrons. Here the analysis is limited to paired halo neutrons to eliminate the complication of the effect of neutron spin pair formation on binding energies. The filled shells may be considered to be composed of rings of modules of neutrons and protons linked together in sequences of the form -n-p-p-n-, or equivaqlently -p-n-n-p-, which may be termed alpha modules because the smallest such unit is the alpha particle.

Such nuclides being considered are effelctively composed of alpha modules and neutron pairs. Here is the plot of the incremental binding energies of nuclides having 14 alpha modules (which means that there are 28 neutrons and 28 protons in the inner shells).

The graph is downward sloping to the right because neutron pairs are repelled from each other. On the other hand if the incremental binding energies of neutron pairs are plotted versus the number of alpha modules, as shown below, the graph is generally upward sloping to the right because neutron pairs and alpha modules are attracted to each other.

The full display is here.

Although the graph is generally rising to the right the values drop for the number of alpha modules corresponding to the conventional magic numbers of 8, 20, 28 and 50.

Below is a display of the incremental binding energies of neutron pairs as a function of the number of neutron pairs beyond those in the alpha modules.

The sharp drops on the right are due to the next neutron shell of 26 to 41 pairs being filled and the subsequent pairs going into the shell beyond 41 neutron pairs. The pattern is regular but it is more comprehensible if the incremental binding energies of the neutron pairs are plotted versus the number of neutron pairs in the outer neutron shell.

Clearly the incremental binding energy is a function of the number of neutron pairs in the neutron shell.

Strictly speaking it is only the cases of for the completely filled shells, #a equal to 1, 4, 10 and 14 that proton pairs are halo; i.e., situated in the next proton shell. For the other values of #a some of the additional proton pairs go into the incompletely filled shells along with the alpha modules in that shell.

Within nuclei neutrons form spin pairs whenever possible. Outside of nuclei such spin pairs do not form. The binding energy due to spin pair formation is on the order of 3 million electron volts (MeV). This means that much of the binding energies shown above is just due to the 3 MeV for the formation of the neutron pair.

The disassociation of a nucleus containing a halo neutron spin pair has to come up with 3 MeV to remove the neutron spin pair and form two separate neutron. For some nuclei this is not possible. Thus energy considerations take precedence over matters of force.

The question then is how could such energy deficient nuclei form. The answer it that they could only have been formed in the Big Bang or the energy rich environment of the interior of a star. Thus the existences of at least some of the halo neutron pair nuclei are accounted for by neutron spin pair formation within a nucleus but not outside.

But more importantly however, when an alpha module goes into the next shell it can be of the form -n-n-p-p-. What can be added to the proton on the right? Nothing other than -n-n-, a neutron pair. Thus the binding energy for the a+1nn case for #a one greater than the filled shell values should be relatively large. Two alpha modules in the next shell could have the form -n-n-pp-…-p-p- and a neutron pair could be attached.

Statistical Results

The binding energy, BE, should be a linear function of the number of alpha modules, the number of neutron pairs and the numbers of the interactions of the three types; alpha modules with alpha modules, alpha modules with neutron pairs and neutron pairs with neutron pairs.

This leads to the following regression equation based upon the 696 cases

BE = 43.00008#a + 12.94634#pp − 0.65348(#a(#a-1)/2)
+ 0.39668(#a#pp) −0.8274(#pp(#pp-1)/2) − 52.57776
[504.9] [93.7] [-113.9]
[47.5] [-54.6] [-68.8]

The coefficient of determination for this regression is 0.9999054. The standard error of the estimate is 4.71 MeV. With an average binding energy of 1121.09 MeV this means that the coefficient of variation for the regression estimates is 0.42 of 1 percent.

The magnitudes of the coefficients for the interaction should be proportional to the product of the strong force charges. Previous work found that if the strong force charge of a proton is taken as 1.0 then the strong force charge of a neutron is −2/3. This means that the strong force charge of an alpha module is (2-4/3)=+2/3 and that of a neutron pair −4/3. Therefore the coefficient of the interactions of alpha modules should be proportional to 4/9 and those of the neutron pairs to 16/9. Because these interaction are repulsions the signs of both should be negative. The interaction of alpha modules and neutron pairs involve an attraction and therefore the sign should be positive. The magnitude of such interactions should be proportional to 8/9.

The predicted signs are borne out by the regression results. The ratio of the coefficient for neutron pair interaction to the coefficient for alpha module interactions with neutorn pairs should be (16/9)/(8/9)=2. The regression results give the value

c_nn/c_an = −0.82748/(0.39668) = −2.0860

It is stunningly close. The ratios of the other coefficients for the interactions of alpha modules and neutron pairs is not nearly so close but they are of the right order of magnitude.

Conclusion

The statistical performance of the module is not perfect but it is remarkably good.

Appendix