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The Cross Differences in Binding Energy as a
Function of the Number of Protons 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.

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.

Although the most prominent feature of the display is the odd-even fluctuation there appears to be more structure to the display. The level and amplitude of the fluctuations appears to change at about 39 proton. Thirty nine is half way through the shell that goes from 29 to 50.

A regression was estimated which was of the form

Δ²n,pB = c0 + c1m + c2e + c3u(p<=39) + c4u*e

where m is the number of protons in the shell (p-28), e equals 1 if p is even and 0 otherwise, u(p≤39) equals 1 if p≤39 and 0 otherwise and u*e is the product of u and e. This latter variable is for capturing any difference in the amplitude of the fluctuations above and below 39 protons.

The regression results are:

Δ²n,pB = −0.07345 + 0.01203m + 0.47477e + 0.32805u(p≤39) + 0.04157u*e
[0.9] [3.8] [1.7] [0.2]

The coefficient of determination for this equation is 0.71260. The t-ratios indicate that only the coefficient for e is significantly diffent for zero at the 95 percent level of confidence.

The regression equation with only e as an explanatory variable is

Δ²n,pB = −0.28816 + 0.48064e
[5.1]

The coefficient of determination however for this equation is only 0.57583.

What comes out of the analysis for this case is that cross difference for nuclides having 50 neutrons is a constant level 2-cycle over the shell with fluctuations of magnitude. 0.48064 MeV.

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

But it is more than just an odd-even fluctuation; the minimum values also exhibit an odd-even fluctuation, indicating there is a cycle of four involved.

A regression of the cross differences on the remainders of the proton number when divided by four has a coefficient of determination of 0.85738. The data for proton number 37 through 50 are in the fourth proton shell; the data for proton number 51 and above are in the fifth proton shell. There could be a shift in the level between the shells. The regression analysis indicates that there is no statistically significant shift in the level between the two shells at the 95 percent level of confidence.

Below are given the cases for the number of neutrons being 70, 80, 90 and 100. In all cases there is no statistically significant dependence of the cross difference on the number of protons in the nuclide.

Below are graphs for the cases of 40, 30, 20 and 10. Here there is an interest effect that does not occur in the previous cases. When the number of protons equals the number of neutrons there is significantly higher level for the cross difference.

For the case of 30 neutrons there is atypical pattern for proton numbers 16 through 20. Above 20 the pattern of the odd-even fluctuations is the same as the other cases.

As in the case of 30 neutrons, the pattern for the lowest proton numbers for the case of 10 neutrons is atypical. Above 9 protons the data fits the usual pattern.

What the displayed cases indicate is the level of the cross difference depends upon the odd-eveness of the proton number and whether the proton number equals the neutron number.


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