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An Exhaustive Display of the Cross
Differences for Nuclear Binding Energy
of a Neutron Pair with a Proton Pair
Starting with the Incremental Binding
Energies of Neutron Pairs

The slope of the relationship between the incremental binding energies of neutron pairs and the number of proton pairs should be equal to the interaction binding energy between the last neutron pair and the last proton pair. That relationships for neutron pairs in the 42 through 55 neutron pair shell and protons in the 42 through 55 proton shell are shown below.

As can be seen the slopes of the various relationships shown are approximately equal. The average slope is 0.845 MeV and thus the average interaction binding energy between a neutron pair and a proton pair is 0.845 MeV.

The interaction binding energy between a neutron pair and a proton pair for the case N=155 is 0.875 MeV.

In the following graphs the headings are the same as in the previous two graphs.

The things to look for are: 1. The linearity of the relationships, 2. The evenness of the spacing of the relationships, particularly their noncrossing..

The wider spacing of the relationships is due to the transition between shells at the magic number of N=126.

The bends in the relationships occur at the transition between proton shells at 41 proton pairs, which corresponds to the magic number of 82 for the number of protons.

The bends at 41 proton pairs are also manifested in the above display.

Here the wider spacing between the relationships is due to the transition between shells at the magic number of neutrons of 82.

The changes in the curvature of the relationships in the above and subsequent displays are due the transition between proton shells at the magic number of 50 protons (25 proton pairs).

The wider spacing of the relationships is again associated with the passage to another shell; in this case at the magic neutron number of 50.

The wider spacing of the relationships cannot in this case be associated with a transition to a higher neutron shell. There are however changes in the curvature of the relationships that are associated with the transition between shells at 14 proton pairs (28 protons).

This display shows a transition between shells for the cases of N=12 and N=14.

As does this one for N=6, 8 and 10. There is a definite pattern here.

Empirical Results

The interaction binding energy of the last neutron and proton pairs is the slope of the relationship shown previously. The value of the slope were computed for the following selection of cases. The method of calculation used is just the ratio of the differences of the end points of the relationships.

Number of
Neutron Pairs
Interaction Binding
Energy of
Neutron and Proton Pairs
10 4.3557
15 3.115
20 2.3726
25 1.9200
30 1.6800
35 1.3800
40 1.1964
45 1.2000
50 1.0910
55 1.1846
60 1.0005
65 1.1828
70 1.0480
75 0.9068
80 0.8850

The number of neutron pairs may be taken as a proxie for the scale of the nucleus. A plot of the interaction binding energy for the above cases versus the number number of neutron pairs is shown below.

There is a definite inverse relationship between the interaction binding energy and the scale of the nucleus or the shell. There is a relationship between the radius of the nucleus R and its number of nucleons A of the form

R = R0A1/3

where R0 is a constant.

The number of neutron pairs N is closely correlated with the total number of nucleons. Some variables are inversely proportional to the square of distance. To look for this type of relationship the interaction binding energies were plotted versus N−2/3. The result is shown below.

This is a remarkable correspondence for the lower values of N.

The force between two particles might be inversely proportional to the square of the separation distance but the potential energy associated with that force would be proportional to the integral of the force, which would make it inversely proportional to the first power of the separation distance. The natural logarithms of the interaction binding energies were regressed on the logarithm of the number of neutron pairs. The result was

ln(IBE) = 3.0693 − 0.73817ln(N)
Coefficient of Determination (R²) = 0.95817

Since the radii of the nuclei are proportional to the cube root of N the coefficient for the dependence on nuclear radius would be − 2.2, not far from −2.

However it is believed that the nuclear strong force involves another term which makes it drop off with distance faster than the reciprocal of distance squared.

Such a force formula is

F = H·exp(−s/σ)/s²

where H and σ are constants.

Let R=N. Then the potential energy for such a force formula might be approximated by a function of the form

IBE = C0exp(−R/σ)/R

A negative exponential function of R has roughly the same shape as the reciprocal function. Therefore IBE would appear to be proportional to 1/R². The question is, given a dependency of IBE on the reciprocal of R, whether the remaining variation is better explained by R or ln(R). This is tested by regressing ln(IBE·R) on R and then on ln(R). The regression of ln(IBE·R) on R has a coefficient of determination of 0.82 while that of ln(IBE·R) on ln(R) has one of 0.87. While that means 1/R² gives a slightly better fit than exp(−R/σ) they are close and the negative exponential function is viable alternative.

The actual relationship, as shown below, of IBE·R versus R is not obviously either of the above alternatives. The relationship for low values of N and thus of R could be linear. However the negative exponential exp(−αR) is approximately (1−αR) for small values of R.


Generally the relationships are approximately linear where there is not a transition between nucleon shells. The relationships do not cross. The interaction binding energies appear to be closely related to the spatial separations within the nuclei, even perhaps obeying an inverse distance squared dependency or a negative exponential weighted inverse distance relationship.

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