San José State University

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Thayer Watkins
Silicon Valley
USA

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

This is a statistical investigation of the incremental binding energies of neutron 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 neutron pair is then

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

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

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

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

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

In other cases 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 nn equals pp. Before that level increases in the the number of neutrons 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 where a, b and c are constants. A quadratic function of two variables x and y takes the form

#### z = a bx +cy +dx² + exy + fy²

with a, b, c, d, e and f being constants.

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 3.70649 34.1
nn -2.31870 -29.0
pp² -0.09009 10.0
nn(pp) 0.07429 6.4
nn² -0.00963 -2.5
pp>14 -0.33730 -0.7
pp>25 -0.22848 -0.6
pp>41 0.55594 -1.2
nn>pp -9.17682 -17.1
nn>14 -2.34179 -4,5
nn>25 -1.57466 -3.7
nn>41 -1.21248 -3.3
nn>63 -3.83226 -6.9
C0 11.87433 24.1

The coefficient of determination (R²) is 0.897, 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 neutron pair. The value for pp is positive indicating the force between a neutron pair and a proton pair is an attraction. The value for nn is negative indicating the force between one neutron pair and another neutron 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 neutron pairs would be proportional to 4q². Thus the ratio of the regression coefficient for nn to that for pp should give the value of q; i.e.,

#### 4q²/4q = q = -2.31870/3.70649 = −0.6255

This is reasonably close to the estimate of q as −2/3 found elsewhere.