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The Distance Dependence
of the Binding Energy
between Neutrons


After Ernest Rutherford discovered that the positive charge of an atom is concentrated in a tiny portion which he called its nucleus (kernel) there was great puzzlement as to how this could be since positive charges repel each other. . Someone realized that if there were a second force that is more attractive at short distances than the electrostatic force is repulsive, but at large distances less attractive an atom's nucleus could be stable.

Ever since then if someone asks any physicist what holds a nucleus together they answer solemly "The Nuclear Strong Force." What this really means is that if such a Nuclear Strong Force exists it would explain how a nucleus of multiple positive charges could be held together. This is an entirely different matter than whether a "Nuclear Strong Force" actually exists.

About 1930 the neutron was discovered. It was soon presumed without any experimental justification that all protons and neutrons were attracted to each other through the hypothetical Nuclear Strong Force. This expanded version of the theory cannot be true because it implies the existence of nuclides with multiple protons but no neutrons. It also implies the existence of nuclides with multiple neutrons and no protons. Furthermore it implies the existence of nuclides with some protons and an unlimited number of neutrons. Such do not exist.

An Alternative Explanation

There is another hypothesis which explains how a nucleus could be held together which involves the spin pairing of nucleons: proton-proton, neutron- neutron and neutron-proton. Spin pairing is strong but exclusive in the sense that one neutron can spin pair with one other neutron and with one proton but no more. The same applies to a proton.

Under this hypothesis there is another much weaker force that prevails between nucleons. It can be called the interactive nucleonic force. It is such that unlike nucleons are attracted to each other but like ones are repelled.

This attraction or repulsion shows up in terms of the incremental binding energies. The incremental binding energy of a neutron for a nuclide is its binding enery less the binding energy of a nuclide having one less neutron. If the incremental binding energy of neutrons decreases as the the number of neutrons in the isotopes increases then it means that neutrons are repelled from one another. It is likewise for protons.

On the other hand, if the incremental binding energy of neutrons increases as the the number of protons in the isomers increases then it means that neutron are attracted to protons. It is likewise for the incremental binding energy of protons.

Incremental Binding Energies

The odd-even fluctuation represent the effect of neutron-neutron spin pairing. The slope of a flat edge of the display represents the force experienced by a neutron.

Determination of the Slopes of the
Flat Edges of the Display
p 12 24 36 48 60
Lower n
IBEn 6.29 8.2564 7.817 12.52 7.38
Upper n
IBEn 3.5 5.9 6.5 6.5 6.41
ΔIBEn −2.79 −2.3564 −1.317 −6.02 −0.97
Δn 6 8 8 30 6
Force =
−0.4650 −0.2946 −0.1646 −0.2007 −0.1617

Determination of the Outer
Radii of the Nuclides
p 12 24 36 48 60
min n 25 41 61 82 101
(min n)+p 37 65 97 130 161
= ((min n)+p)1/3
3.3321 4.0202 4.5940 5.065 5.4392

The force between particles is proportional to the second difference of the binding energy. That is equivalent to the slope of the relationship ofincremental binding energy and the number of particles of a type in the nuclides.

The distance dependence of the forces between particles which can be measured; the electrostatic, the gravitational and the magnetic; are all proportional to the reciprocal to separation distance squared. The material here is to investigate the distance dependence of the interaction force between neutrons.

If the force between nucleons has the distance dependence of

F = αD−β


log(F) = log(α) − βlog(D)

Thus the exponent β can be found by regressing log(F) on log(D).

That regression coefficient is −2.1533; i.e. approximately −2. The standard deviation of log(D) is 0.08465. The ratio of the deviation of the regression coefficient from −2 to this standard deviation is −1.811. Thus the regression coefficient is is not statistically different from −2 at the 95 percent level of confidence.


The distance dependence of the interactive nucleonic force between neutrons in nuclei is inverse distance squared.

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