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The Statistical Explanation of the
Incremental Binding Energies
of Proton Spin Pairs

This is a statistical investigation of the incremental binding energies of proton spin pairs. In order to keep the analysis from being complicated by the binding energies due to the formation of spin pairs only the nuclides which consist entirely of spin pairs are considered. There are 738 such nuclides and the incremental binding energy for 657 can be computed.

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 neutron and pp proton 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.

When the incremental binding energies for proton pairs are examined it is found that often they are nearly linear, such as for the case seen below.

In other case there is a drop in the level after a proton shell is filled, as is shown below

A drop in the level may also occur after pp equals nn. Before that level an increase in the protons results in the formation of neutron-proton spin pairs.

A quadratic function of a single variable x allows for a dependence of the form z=a+bx+cx². A quadratic function of two variables x and y takes the form

z = a bx +cy +dx² + exy + fy²

In the analysis a variable of the form denoted as nn>N means that the variable is 1 if nn is greater than N and zero otherwise.

Regression Coefficients and their t-Ratios
Variable Coefficient t-Ratio
pp -2.36603 -40.7
nn 1.62516 38.4
pp² 0.03262 14.6
nn(pp) -0.03311 -11.6
0.00675 7.1
pp>14 -2.0097 -4.5
pp>25 -3.30886 -9.8
pp>41 -2.80764 -6.6
pp>nn -11.21996 -26.1
nn>14 -1.89566 -3.9
nn>25 -1.87753 -4.8
nn>41 -1.87795 -5.5
nn>63 1.72115 3.3
C0 30.60738 61.8

The coefficient of determination (R²) is 0.959, good but not spectacular. The most interesting thing in the results is the comparison of the coefficients for nn and pp. They represent the interaction of a neutron pair and a proton pair, respectively, with a proton pair. The value for nn is positive indicating the force between a neutron pair and a proton pair is an attraction. The value for pp is negative indicating the force between one proton pair and another proton pair is a repulsion.

If the nuclear strong force charges of a proton and a neutron are denoted as 1 and q, respectively, then a proton pair has a strong force charge of +2 and a neutron pair a charge of 2q. Thus the interaction between a proton pair and a neutron pair would be proportional to 4q and between two proton pairs would be proportional to 4. Thus the ratio of the regression coefficient for nn to that for pp should give the value of q; i.e.,

4q/4 = q = 1.62516/(-2.36603) = −0.68687

This is entirely consistent with the estimate of q as −2/3 found elsewhere.


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