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 pair formation the analysis is limited to the nuclides that consist entirely of spin pairs. There are 728 such nuclides and for 628 of them the incremental binding energy can be computed..

Let nn and pp be the numbers of neutron and proton 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

Δ_nnB²E(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 neutron shell.

The first differences in the binding energies of the isotopes of Krypton with an even number of neutrons are shown below..

The relationship appears to be linear except for a drop in the level after nn=25. Twenty five neutron pairs corresponds to 50 neutrons, one of the nuclear magic numbers representing filled shells. The slope of the relationship after nn=25 may or may not be the same as the slope of the relationship up to nn=25. This can be tested by fitting to the data a regression function of the form

IBE(nn) = c₀ + c₁nn + c₂i(nn-25) + c₃u(nn-25)

where i(z)=0 if z≤0 and i(z)=1 if z>0 and u(z)=0 if z≤ and u(z)=z if z>0.

Empirical Results

In this preliminary analysis only the isotopes of Krypton will be considered. Here are the data for these nuclides.

The regression results are:

IBE = 47.05542 − 1.21596nn −1.17868i(nn-25) −[0.57857u(nn-25)
[13.5]         [-7.6     ]             [-1.9]]              [-2.1]

The numbers in square brackets, [ ], at the t-ratios for the coefficients above them. The t-ratio for a regression coefficient is the ratio of its value to the standard deviation of its estimate. For a regression to be significantly different for zero at the 95 percent of confidence its t-ratio must be greater than about 2 in magnitude.

The regression results indicate the drop in value after 25 neutron pairs and the difference in the slopes of the relationships below and above 25 neutron pairs are of marginal statistical significance.

The coefficient of determination (R²) for the regression is 0.988.

Second Differences

The graph of second diffences reveals an interesting phenomenon.

With the shell that ranges from 29 neutrons to 50 neutrons the values of the second differences appear to be sinusoidal. The period of the cycle appears to be 6 neutron pairs. An appropriate regression equation is

Δ_nn²BE(nn, pp) = c₀ + c₁cos((2π/6)(nn-21) + c₂sin((2π/6)(nn-21))

The results the regression of the second differences from nn=20 to nn=30 are

Δ_nn²BE(nn, 18) = −1.20033 −0.36417cos((2π/6)(nn-21)) − 0.07304sin((2π/6)(nn-21))
[-37.4]               [-8.1     ]                  [-1.6]

The coefficient of determination (R²) for this regression is 0.957.

The binding energy due the interaction of the last two neutron pairs is from the above −1,20033 MeV.The interaction of two neutron pairs involve four neutron-neutron interactions. Therefore the interaction of two neutrons is −0.30008 MeV, a repulsion.

(To be continued.)