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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 Proton Pairs

The slope of the relationship between the incremental binding energies of proton pairs and the number of neutron pairs should be equal to the interaction binding energy between the last proton pair and the last neutron pair. Those relationships for neutron pairs in the 72 through 80 range and protons in the 101 through 105 proton range are shown below.

As can be seen the slopes of the various relationships shown are approximately equal. The average slope is 0.906 MeV and thus the average interaction binding energy between a proton pair and a neutron pair is 0.906 MeV.

The interaction binding energy between a neutron pair and a proton pair for the case N=100 is 0.884 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 linearity indicates the interaction energies are the same for all pairs in a shell.

The bends in the relationships occur at the transition between neutron shells at 63 proton pairs, which corresponds to the magic number of 126 for the number of neutrons.

The lower two relationships also display a bend at 63 neutron pairs. The widening of differences between the relationships is due to a transition between proton shells at the magic proton number of 82 (41 proton pairs).

There is a kink in the relationships at 41 neutron pairs (82 neutrons).

The wider separation is associated with the transition in proton shells at 50 protons. The kink associated with 41 neutron pairs also shows up in the above display.

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.

The Interaction Binding Energy
between a Neutron Pair
and a Proton Pair
of Neutrons
in Nuclide
Binding Energy
20 3.7280
25 3.210
30 2.6285
35 2.3767
40 1.778
45 1.7143
50 1.5129
55 1.2594
60 1.2750
65 1.2133
70 1.1288
75 1.0601
80 1.1209
85 0.8750
90 1.0046
95 0.9524
100 0.8838
105 0.9060

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

This is a remarkable correspondence. 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 neutrons is closely correlated with the total number of nucleons. Thus N is a proxy for the radius of the nucleus.

To see why such a relationship should exist between the interaction binding energy and the radius consider the special case of two particles in which the force acting on each is inversely proportional to the square of the distance of separation. If the charge of one is distributed uniformly over a spherical shell then the force between them would be the same as if the charge were concentrated at the center of the spherical shell. If the center of the shell is the center of the nucleus then the force would be inversely proportional to the square of the distance of one particle from the center of the nucleus.. The potential energy would be proportional to the reciprocal of the distance of a particle from the center of the nucleus.

The particles' charges would not have to be uniformly distributed over a spherical shell. If their trajectories took them all over a spherical shell the time average of the force and the potential energy would be the same as if the charges were so distributed.

For forces that do have not strictly inverse distance squared dependence the relationships are not so simple but there would still be a dependence upon the distance of the outer particle from the center of the nucleus.

When the logarithm of the interactive binding energy is regressed on the logarithm of N, 97.25 percent of the variation in the logarithm of IBE is explained by the variation in the logarithm of N.

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.

The potential energy for such a force formula can be approximated by a function of the form

IBE = C0exp(−R/σ)/R

This can be tested by regressing the logarithm of the product of IBE and R on R. That regression shows that 90.55 percent of the variation in the dependent variable is explained by variation in R. Somewhat more, 93.37 percent, is explained if the logarithm of R is used as the explanatory variable.


Generally the relationships are approximately linear where there is not a transition between nucleon shells. The relationships do not cross. There is an inverse relationship between the interaction binding energy of a neutron and a proton and the scale of the nucleus. The results are consistent with the formula for the strong force having a negative exponential weighted inverse distance square dependency.

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