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Tests of the Equality of the Cross
Differences of Nuclear Binding Energies

An argument was developed elsewhere that the cross difference of the binding energies of neutrons and protons gives the binding energy due to the interaction of the last neutron with the last proton. The cross differences can be calculated two ways but should give the same value. This webpage is an attempt to test this proposition with the data on binding energies. The actual binding energies are more complicated than the proposition allows. Binding energies not only reflect the interactions through the so-called nuclear strong force but are also affected by the formation of spin pairs. Furthermore the two neutrons and two protons can form a module of the nature -p-n-n-p-, or equivalently, -n-p-p-n-. The simplest form of such a module is the alpha particle. Hereafter these modules will be referred to as alpha modules.

The influence of spin pair formation can be eliminated by considering only nuclides made up entirely of spin pairs and taking the units of analysis to be spin pairs. However the incremental differences are affected by the formation of alpha modules and this limits the validity of the proposition concerning cross differences.

Rather than dealing with the cross differences per se it is convenient to view the first differences of binding energy with respect to the number of neutron pairs plotted versus the number of proton pairs and compare this with the first differences with respect to the number of proton pairs plotted versus the number of neutron pairs. The slopes of these relationships are the cross differences. If the plots are parallel curves then the proposition holds. Here are the plots.

The above curves do indeed appear to be parallel even though there are significantly larger jumps occurring in them. In the next three the significantly larger jump occurs at different points and this destroys the strict parallelism of the curves, but aside from this the curves do appear to be merely shifted versions of the same curve.

The plots are similar except that one curve seems to be shifted one unit with respect to the other one. Here is a graph with one curve shifted a unit with respect to the other.

The graphs for nuclides with somewhat higher numbers of neutron and proton spin pairs look essentially the same as the one shown previously.

In the graph below the plots are obviously not parallel but they could be said to be "roughly" parallel.

However the "roughly" is quite rough. The slope of the IBEPP line is 74.9 percent of the slope of the IBENN line.

For the case below the plots do not overlap so no comparison of the slopes can be made, nor can they be described as "roughly" parallel. Howerver if they are extrapolated into the range of the other a comparison can be made and they can be said to be "roughtly" parallel.. Again however the "roughly" is quite rough. The slope of the IBEPP line is 75.9 percent of that of the IBENN line. The ratio is notably close to the 0.749 ratio found in the previous case. There is obviously a story to tell here that is different from the one being pursued.


The proposition seems to hold for smaller nuclides but not for the larger nuclides. But even for the larger nuclides the slopes and hence the cross differences are "roughly" equal, being of the same order of magnitude. For the larger nuclides the slope of the IBEPP curve with respect to the number of neutrons is about three quarters of the corresponding slope of the IBENN line with respect to the number of protons. According to the analysis each of the slopes should be equal to the binding energy associated with the interaction of the last neutron with the last proton.

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