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The Statistical Testing of the Cross Differences
in Binding Energies for Dependence on the
Number of Nucleons in the Nucleonic Shells

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 49, 50 and 51 protons as a function of the number of neutrons in nuclide. The values are shown only for the odd number of neutrons in the nuclides to avoid the effect of neutron spin pairing. The odd numbered values show the interaction solely due to the nuclear strong force.

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 regressions were run using only the data for neutrons in the sixth shell and then only for the 55 through 81 neutron cases. The cases of 51 and 53 clearly do not fit the pattern for the other cases in the sixth shell.

The t-ratios (the regression coefficients divided by their standard deviations) are −0.7, −1.5 and −2.3 for protons number 49, 50 and 51, respectively. Thus the regression coefficients for p=49 and p=50 are not significantly different from zero at the 95 percent level of confidence. The one for p=51 is significant, but barely so, at the 95 percent level of confidence.

The fact that the general levels of the p=49 and p=51 are about the same but the general level of the p=50 indicates a possible odd-even fluctuation with respect to the proton number.

Below is given the graph for proton numbers 59, 61 and 63.

Here the general levels for the three odd proton numbers are the same.

For the case of proton numbers 69, 71 and 73, shown in the graph below, the general level seem to be the same but there appears to be a shift in the level for neutron number 91.

A shift at about neutron number 91 indicates that something happens after there are eight neutrons in the shell.

Examples of the pattern for the cross differences as a function of the number of protons in the nuclide are shown in the diagram below.

The pattern that emerges is that shown below in which the cross differences are roughly constant for the midranges of nucleons in a shell but divergent for the first few nucleons in a shell and to a lesser extent or not at all for the last few nucleons in the shell.

The testing for the constancy of the cross differences in binding energy in the midrange of shell occupancy could be carried out for all the odd proton number and odd neutron number cases but there are too many of them. The EXCEL regression program limits the number of parameters to be computed to 16. Therefore it is necessary to limit the cases considered. The cases considered were the interaction of a proton in the sixth shell with neutrons in the sixth and seventh shells, a proton in the seventh shell with neutrons in the seventh and eighth shells. There are 396 nuclides of these types.

What is needed is a regression in which all 396 cases are utilized and the cross difference for n neutrons and p protons is a linear function of the number of neutrons in the neutron shell corresponding to n and the number of protons in the proton shell corresponding to p. The desired outcome of the regression is that the regression coefficients for the numbers of nucleons in the shells are not significantly different from zero.

(To be continued.)

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