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The Binding Energies of Nuclides
with Halo Proton Pairs

Halo Nucleons

Halo nucleons occur in nuclides which otherwise have filled shells of nucleons. It is easy to see why halo neutrons should exist. The filled shells consist of equal numbers of neutrons and protons. Neutrons are attracted through the so-called nuclear strong force to protons but repelled by other neutrons. However the strong-force charge of the protons is greater than that of neutrons so neutrons on balance are attracted to a core nucleus of equal numbers of protons and neutrons.

Since in nuclei whenever possible neutrons and protons form spin pairs the core nucleus of equal numbers of protons and neutrons consist of shells of chains of nucleons of the form -n-p-p-n-, or equivalently -p-n-n-p-. Such sequence can be appropriately called alpha modules because the smallest such unit is an alpha particle. The nuclides which consist entirely of alpha modules are referred to as alpha nuclides.

There are a great number of nuclides with halo neutrons. There are relatively few with halo protons. The problem is that it would seem that there should not be any. Shown below are the cases involving proton pairs. The analysis is limited to to proton pairs to eliminate the distraction of the odd-even fluctuation in incremental binding energy due to the nonformation/formation of proton spin pairs. The incremental binding energy of a unit is the binding of a nuclide less the binding energy of the nucide having one fewer of the units.

The incremental binding energy of a proton pair includes that due to the formation of a proton spin pair and any adjustments in the arrangement of the other nucleons as a result of the addition of a proton pair to the nucleus. In addition to these sources of binding energy there is the binding energy due to the interaction of an additional proton pair with the other nucleons, including other halo proton pairs as well as those in the alpha modules of the core nucleus.

Here are the values of the incremental binding energies of proton pairs where the notation a+kpp refers to a nuclide which consists of an alpha nuclide plus k proton pairs. The term #a stands for the number of alpha modules in a core nucleus.

The Incremental Binding Energies of Proton
Pairs in Nuclides that are Otherwise Composed
Entirely of Alpha Modules
(MeV)
#a a+1pp a+2pp a+3pp
1 -1.371674 -2.142
2 3.82099 -1.7715
3 6.571502 -1.40823
4 4.534164 2.3165 -0.02
5 7.932741 3.4264 -0.604
6 7.78911 3.364 -1.41
7 7.14811 2.695 -0.78
8 6.94034 2.639 -0.96
9 6.4063 1.368 -0.29
10 4.853 2.995 0.3
11 6.5003 3.215 0.31
12 6.238 2.76 -1.56
13 5.453 1.15 -2.8
14 2.972 0.05
15 2.638
16 1.85
17 1.4
18 1.1

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.

The explanation of the existence of halo proton pair nuclei has to be in terms of energy rather than forces because of the importance of spin pair formation. Within nuclei protons 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 proton pair.

The disassociation of a nucleus containing a halo proton spin pair has to come up with 3 MeV to remove the proton spin pair and form two separate protons. 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 proton pair nuclei are accounted for by proton 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 --p-p-n-n-. What can be added to the neutron on the right? Nothing other than -p-p-, a proton pair. Thus the binding energy for the a+1pp 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-p-p-n-p-p-n-n- and a proton pair coulld be attached. Two alpha modules and two proton pairs could form two chains of nucleons. Thus while a proton pair may be repelled from alpha modules through the strong force the formation of spin pairs may overcome that repulsion within limits. The repulsion of too many nearby alpha modules may overwhelm the effect of such formations of nucleon chains in an outer shell.

Statistical Results

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

This leads to the following regression equation based upon the 67 cases

BE = 37.61168#a + 3.63975#pp − 0.33107(#a(#a-1)/2)
+ 0.12548(#a#pp) −2.49807(#pp(#pp-1)/2) − 18.36308
[161.9] [3.8] [-18.2]
[1.6] [-3.9] [-13.2]

The coefficient of determination for this regression is 0.99987. The standard error of the estimate is 2.45 MeV. With an average binding energy of 348.75 MeV this means that the coefficient of variation for the regression estimates is 0.7 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 proton pair +2. Therefore the coefficient of the interactions of alpha modules should be proportional to 4/9 and those of the proton pairs to 4. Because these interaction are repulsions the signs of both should be negative. The interaction of alpha modules and proton pairs involve and interaction and therefore the sign should be positive. The magnitude of such interactions should be proportional to 4/3.

The predicted signs are borne out by the regression results. The ratio of the coefficient for proton pair pair interaction to the coefficient for alpha module interactions should be 4/(4/9)=9. The regression results give the value

cpp/caa = −2.49807/(−0.33107) = 7.55

It is not 9 but it is the right order of magnitude. The coefficient for the interactions of alpha modules and proton pairs is not of the right order of magnitude and is is not statistically significantly different from zero at the 95 percent level of confidence.

Conclusion

Although the proton pairs are repelled by the alpha modules in filled shells the possibilities of the formation of nucleon chains by alpha modules and proton pairs in an outer shell may overcome that repulsion and account for the existence of nuclides consisting of alpha modules and halo proton pairs.

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


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