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The Binding Energy of Nuclei Due
to the Interaction of Proton Pairs

There is a theorem that says that the second difference in binding energies for a particle gives the interaction binding energy of the last one of those particles with the next to last one. In order to keep the results from being affected by the odd-even fluctuations due to spin pairs only nuclides with even numbers of protons and neutrons will be considered. There are 728 such nuclei and for 657 the incremental binding energy of a proton pair can be calculated.

Let pp and nn be the numbers of proton and neutron pairs, respectively. Let BE(nn, pp) be the binding energy of the nuclide with nn proton and pp neutron pairs. The incremental binding energy of a proton pair is then

IBE(nn, pp) = BE(nn, pp) − BE(nn, pp-1)

The second difference in binding energy with respect to the number of proton pairs is then

ΔppBE(nn, pp) = IBE(nn, pp) − IBE(nn, pp-1)
= BE(nn, pp) − 2BE(nn, pp-1) + BE(nn, pp-2)

For the second difference to measure the interaction of the last two proton pairs, pp, pp-1 and pp-2 must be in the same proton shell.

The analysis begins with the first differences in the binding energies of the isotopes of heavier elements with an even number of protons. There is more regularity for such elements than for the lighter elements. First take the case of the isomers with nn=94 shown below..

3328671 .

The relationship appears to be linear and the variation in the number of proton spin pairs explains 99.5 per3cent of the variation in the incremental binding energy. The slope of the relationship, based upon regression analysis, is −2.1533 MeV per proton pair per proton pair. This is −0.53833 MeV per proton-proton interaction.

However despite the apparent linearity of the above relationship the second differences reveal a pattern of deviations from linearity.

This relationship appears to be approximately linear. There is a slight upward trend in the values. Such a function explains 78.6 percent of the variation in the second differences.

The integral of a linear function is a quadratic function so a quadratic function may be included in the regression equation for the incremental binding energies of proton pairs. When this is carried out the coefficient of determination (R²) for the regression is raised from 0.9952 to 0.9997.

Linear deviations from constancy are often an element of the relationship between incremental binding energy and the number of proton pairs. Take the case of the nuclides with 47 proton pairs.

The relationship of incremental binding energies and the number of proton pairs is very regular and almost a straight line. A linear regression equation explains 99.575 percent of the variation in the incremental binding energies of the proton pairs in nuclides with 47 neutron pairs. A linear regression equation explains 56.2 percent of the variation in the second differences. There a regression with linear and quadratic terms should explain even more than 99.575 percent of the variation in the incremental binding energies of the proton pairs. And it does: 99.96 percent.

Now consider the case of nn=42.

The slope is negative and becomes slightly less negative. A quadratic regression equation explains 99.96 percent of the variation in the incremental binding energy of the proton pairs.

For the nuclides with nn=36 there are two shells involved divided by pp=25, which corresponds to the filled shell magic number of 50.

A quadratic equation with a drop in the level after pp=25 explains 99.976 percent of the variation in the incremental binding energies of the proton pairs; The drop after pp=25 is 2.22 MeV. The coefficient of pp² is not statistically significantly different from zero at the 95 percent level of confidence. A linear function with a drop after pp=25 explains 99.974 percent of the variation.

For the nuclides with nn=30 the slope is negative but with a drop in the level after pp=25.

For nuclides with 24 neutron spin pairs the incremental binding energy of proton spin pairs decreases with increasing numbers of proton spin pairs but at a slightly decreasing rate.

For nuclides with 18 neutron spin pairs the graph is separated into two downward sloping lines separated at 14 proton spin pairs which corresponds to a filled shell magic number of 28.

There also a drop in the level after 18 proton spin pairs because that is where pp is equal to nn, and thus afterwards there are no neutron-proton spin pairs formed.

For the case of nn=12 the relationship is downward sloping but irregular. There is a sharpe drop after pp=12 because that is where pp equals nn and thus above that level no neutron-proton spin pairs are formed.

(To be continued.) .


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