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The Separation of the Cross Differences in Binding Energy
into the Component Due to Pair Formation and the Component
Due to the Interaction Through the Nuclear Strong Force

Let B(n, p) be the binding energy of a nuclide with n neutrons and p protons. The incremental binding energy of the last neutron in such a nuclide is

ΔnB(n, p) = B(n, p) − B(n-1, p)

The cross difference is the increment in the incremental binding energy of the neutron due to change in the number of protons in the nuclide; i.e.,

Δ²n,pB(n, p) = ΔnB(n, p) − ΔnB(n, p-1)
which is equivalent to
Δ²n,pB(n, p) = B(n, p) − B(n-1, p) + B(n-1, p-1) − B(n-1, p)

A little algebraic manipulation shows that

Δ²n,pB(n, p) = Δ²p, nB(n, p)

In other words, the two cross differences are equal.

Nuclear Shells

Protons and neutrons are arranged separately in shells. The numbers corresponding to the shells filled to full capacity are known as the nuclear magic numbers. Conventionally the magic numbers are {2, 8, 20, 28, 50, 82, 126}, but a case can be made for the magic numbers being instead {2, 6, 14, 28, 50, 82, 126} with 8 and 20 being in a different category of magic numbers. For more on this see Magic Numbers.

The structure of the nuclear shells, both for neutrons and protons, is given in the following table.

Range1 to 23 to 67 to 1415 to 2829 to 5051 to 8283 to 126127 to 184

The Cross Difference as the Interaction Energy
of the n-th Neutron with the p-th Proton

The case is made Elsewhere that the cross difference for a nuclide of n neutrons and p protons is the interaction energy of the n-th neutron with the p-th proton. Here are the cross differences for the nuclides with 50 neutrons as a function of the number of protons in nuclide.

This display clearly shows that a pairing phenomenon is involved. The interaction energy of a neutron with a proton involves both the interaction through the strong nuclear force and a spin pair formation. However by separating the data for even numbers of protons from that of odd number of protons the effect strictly of the strong force can be analyzed.


There is reason to expect the cross differences to be the same for all protons in the same shell. This would be manifested as a constant level with random variations about that constant level. That would mean that the regression coefficient of the cross differences on the number of protons would not be significantly different from zero.

The t-ratio (the regression coefficient divided by its standard deviation for the even case above) is −0.5 and thus the regression coefficient is not significantly different from zero at the 95 percent level of confidence. The average energy for the even case is 0.71843 MeV. This of course includes the effect of pair formation as weell as the average interaction energy. The standard deviation of the data points is 0.25562 MeV.

For the odd case the t-ratio is −2.5 and thus it is significantly different from zero at the 95 percent level of confience. Bu the regression is highly influenced by the first two data points, the ones for 29 and 31 protons. These are near the transition between shell levels. If those two data points are left out of the regression the t-ration is −0.5. The average interaction energy excluding the first two data points is 0.17063 MeV. The standard deviation of these data points is 0.09800 MeV. These figures are for the interaction of a neutron in the fifth neutron shell with protons in the fifth proton shell.

The data for the case for the 60th neutron versus the number of proton also exhibits the odd-even fluctuation.

The separate even and odd proton number cases are shown below.


In neither case are the regression coefficients for the number of protons statistically significant at the 95 percent level of confidence. The sixtieth neutron is in the sixth neutron shell whereas the fiftieth and below protons are in the fifth proton shell. The average and the standard deviation for the odd number case are 0.06566 MeV and 0.11510 MeV, respectively. The average is notably smaller than the case for the fiftieth neuton, 0.17063 MeV, which is the interaction of a neutron in the fifth neutron shell with protons in the fifth proton shell.

The cases for the seventieth neutron are a bit different from the previous examples.


Likewise for the case of the eightieth neutron:


There appears to be a cycle for the even case but that is not the case for nearby cases.

(To be continued.)

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