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The Binding Energies of Mirror Nuclides

The binding energy of H3 (tritium) with one proton and two neutrons is 8.48182 MeV whereas that of He3 with two protons and one neutron is 7.718058 Mev. The difference 0.763763 MeV is due to the electrostatic repulsion of the two protons in He3. This value provides information on the separation distance of the two protons in He3. The 0.764 Mev is 1.224×10−13 joules. The coulomb potential for two protons is 2.31×10−28/d, d is the distance of separation. Thus

d = (2.31×10−28)/(1.224×10−13) = (1.887×10−15) = 1.887 fermi

The working hypothesis is that the separation distances of the nucleons are the same in nuclides having the same number of nucleons but differing in the number of proton-proton interactions.

The data for the change in binding energies for the (P, N) nuclides compared to the mirror nuclide of (N, P). From this set the cases for which the proton number decreased exactly one unit were compiled. These values are plotted in the graph below.

The effect on binding energy increases with the number of protons in the nuclide because the eliminated proton had been interacting with all of the other protons. There is a slight curvature indicating that the interaction is less strong for a larger number of protons because there is a greater separation. A careful viewing of the graph reveals that the relationship rises slightly after four protons and then drops back after 20 protons.

A regression equation of a quadratic form with changes at the P=5 and P=20 levels was estimated. The result was

ΔBE = 0.345192 + 0.404675P − 0.00297P² + 0.345560d(P-5) − 0.16973d(P-20)
[2.12] [29.0] [-13.2] [2.56] -[1.52]
R² = 0.998111

The correlation between the data and the regression estimates is quite close. The following graph has the data and the regression estimates plotted together. The regression estimates are on the upper edge of the yellow area.

The only discernable differences are in the upper range of the proton number from 30 to 35.

The increments in the values for an increase in the number of protons in the nuclide give greater detail, as shown below.

The increments can be computed from the regression equation. These are plotted along with the actual increments in the graph below.

The regression equation provides an estimate of the illusive binding energy of a neutron pair. The first data point is the difference between the binding energy of tritium H3 and the He3 isotope. This shows up as the value for P=1. The value for P=0 would be the difference between the binding energy of neutron pair and the binding energy of the deuteron. Thus

binding energy of a neutron pair = 0.345192 + 2.224573 = 2.569765 MeV

This value is not consistent with the data for the binding energy of small nuclides. For example, the binding energy of He6, the nuclide with two neutrons more than He is 29.2691 MeV, less than 1 MeV greater than that of He4. The data for the other nuclides having two more neutrons than the alpha nuclides are displayed below.

The same approach can be used to estimate the binding energy of a neutron quartet. First, the effect of switching two protons to neutrons is compiled. The result is displayed below.

There does not appear to be shifts at the magic number points so a simple quadratic regression is used; i.e.,

ΔBE = 1.376356 + 0.82649P − 0.00635P²
[5.15] [42.7] [-12.82]
R² = 0.998153

The intercept value corresponds to a value of P=0, which means converting the two protons in He4 (alpha particle) to neutrons. This would be a neutron quartet. Thus

binding energy of neutron quartet = 1.376356 + 28.295674 = 29.67203

Again this estimate is not consistent with the data for small nuclides. The binding energy of He8, the nuclide that has four more neutrons than He4 is 31.408 MeV, only 3.11 MeV more than that of He4. The data for the all of the nuclides having four more neutrons than the alpha nuclides are shown below.

The difference between the effect of changing two protons to neutrons and the effect of changing only on proton is intriguing.

The effect of switching three protons to neutrons looks much the same as the one for switching two protons.

A notable aspect of this relations is that the point where there is a shift from one quadratic function to another is where the number of protons is equal to 20, a magic number. The change at 20 protons shows up more clearly in the graph of the increments in binding energy; i.e.,

The difference in the effect of switching three protons compared to switching two protons does not display this remarkable pattern shown previously of the effect of switching two compared to switching one.

It is notable that the pattern changes after 6 protons and after 28 protons, both magic numbers. The change after 28 is ambiguous, but the pattern that clearly prevails after 30 protons can be extended backwards to incorporate the data point for 29 protons.

It could be that remarkable pattern for the case of switching two protons compared to switching just one shows up only when a proton pair is eliminated and a neutron pair created. For this reason the data for switching four protons into neutrons was compiled. The results are shown below.

(To be continued.)

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