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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
as a Function of the Number of Neutrons in the Nuclide

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 protons as a function of the number of neutrons in nuclide.


There is reason to expect the cross differences to be the same for all neutrons 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 neutrons would not be significantly different from zero.

The regression was run using only the data for neutrons in the sixth shell (52 through 82 neutrons). The t-ratio (the regression coefficient divided by its standard deviation for the even case above) is 0.995 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 0.676 MeV. This of course includes the effect of pair formation as well as the average interaction energy. The standard deviation of the data points is 0.225 MeV.

For the odd case the t-ratio is 0.57 and thus it is not significantly different from zero at the 95 percent level of confience. The average interaction energy is 0.0991 MeV. The standard deviation of these data points is 0.225 MeV. These figures are for the interaction of a proton in the fifth proton shell with neutrons in the fifth neutron shell.

(To be continued.)

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