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Neutrons repel each other through the nuclear
strong force, as do protons.      Nuclei are held
together by the formation of nucleon spin pairs
and the attraction between neutrons and protons.

Background

Conventional nuclear theory holds that there is a force that involves an attraction between all nucleons (neutrons and protons). This hypothetical force hypothetically drops off with separation distance faster than inverse distance squared and therefore at small separation distances can be stronger than the electrostatic repulsion between protons but at larger separation distances can be weaker. This hypothetical force is therefore called the nuclear strong force. There is no more evidence for its existence than that nuclei containing protons hold together. This theory leaves out the phenomenon of spin pair formation among nucleons and it is this spin pair formation which dominates nuclear structure. Spin pair formation is exclusive in the sense that one neutron can form a spin pair with one other neutron and with one proton. The same applies to protons. Because of this exclusivity spin pair formation does not involve a field in the way the electrostatic interaction of charged particles does. There is however a force field involving the nonexclusive interaction of nucleons.

This is an extreme important topic and therefore the argument will be laid in explicit detail but in such a way that the reader can easily skip over the parts that he or she have no questions concerning.

Thus neutrons repel each other but are attracted to protons. What follows is an introduction to the more complete model of nuclear structure. de



Second Differences

Cross Differences

The increase with respect to the number of protons in
the incremental binding energies of neutrons is equal to the
interaction of the last neutron with the last proton.

Proof:
Consider a nuclide with n neutrons and p protons. The binding energy of that nuclide represents the net sum of the interactions of all n neutrons with each other, all p protons with each other and all np interactions of neutrons with protons.

   

The black squares indicate there are not any interactions of a nucleon with itself.

The neutron incremental binding energy is the difference in the binding energy of the nuclide with n neutrons and p protons and that of the nuclide with n-1 neutrons and p protons. In the diagrams below the interactions of the nuclide with (n-1) neutrons and p protons are shown in color.

   

The subtraction eliminates all the interactions of the p protons with each other. It also eliminates the interactions of the n-1 neutrons with each other and the n-1 neutrons with the p protons. What are left are the interactions of the n-th neutron with the other n-1 neutrons and the interaction of the n-th neutron with the p-th proton.

  

Now consider the difference of the IBE for n neutrons and p protons and the IBE for n neutrons and p-1 protons. In the diagrams below the interactions for the IBE for the nuclide with (p-1) protons are shown colored.

  

The subtraction eliminates the interactions of the n-th neutron with the other (n-1) neutrons. It also eliminates the interactions of the n-th with the (p-1) protons. What is left is the interaction of the n-th neutron with the p-th proton.

Strict Second Differences

The increase in the incremental binding energies of a neutron pair as a result of an increase in the number of neutron pairs is equal to the interaction of the last neutron pair with the next to last neutron pair, provided these two are in the same neutron shell.

Rationale:
Consider a nuclide with n neutron pairs and p proton pairs. The binding energy of that nuclide represents the net sum of the interaction energies of all n neutron pairs with each other, all p proton pairs with each other and all np interactions of neutron pairs with proton pairs. Below is a schematic depiction of the interactions.

  

The black squares are to indicate that there is no interaction of a neutron pair with itself. The diagram might seem to suggest a double counting of the interactions but that is not the case.

The incremental binding energy of a neutron pair is the difference in the binding energy of the nuclide with n neutron pairs and p proton pairs and that of the nuclide with n-1 neutron pairs and p proton pairs. In the diagrams below the interactions for the nuclide with (n-1) neutron pairs and p proton pairs are colored.

  

That subtraction eliminates all the interactions of the p proton pairs with each other. It also eliminates the interactions of the (n-1) proton pairs with each other and the n-1 neutron pairs with the p proton pairs. What are left are the interactions of the n-th neutron pair with the other (n-1) neutrons and the interaction of the n-th neutron pair with the p proton pairs.

Now consider the difference of the IBE for n neutron pairs and p proton pairs and the IBE for (n-1) neutron pairs and p proton pairs. These are shown as the white squares in the diagrams below. The colored squares are the interactions for the IBE of a neutron pair in a nuclide of (n-1) neutron pairs and p proton pairs.

 

The subtraction of the IBE for (n-1) neutron pairs and p proton pairs from the IBE for n neutron pairs and p proton pairs depends upon the magnitude of the interaction of the (n-1)-th neutron pair with the different neutron pairs compared to the interaction of the n-th neutron pair with those same neutron pairs. Visually this is the subtraction the values in the green squares from the white squares on the same level. When the n-th and the (n-1)-th neutron pairs are in the same shell the magnitudes of the interactions with any proton pair are, to the first order of approximation, equal. This is from the previous analysis concerning cross differences. Thus the interactions with the p proton pairs are entirely eliminated.

It would be expected that the constancy of the magnitude of the interactions of neutron pairs and proton pairs for neutron pairs within the same shell would apply also to interactions of neutron pairs with other neutron pairs. In that case the interactions of the n-th and (n-1)-th neutron pairs with the first (n-2) neutron pairs are also eliminated. All that is left then is the interaction of the n-th neutron pair with the (n-1)-th neutron pair.

However if there is any doubt as the equality of the interaction of the k-th and (k-1)th neutron pair and that of the interaction of the (k-1) and the (k-2)-th neutron pair then it should be noted that the second difference is an upper limit for the interaction of the last two neutron pairs and since the second difference is negative the interaction would be more negative.

For the corresponding analysis for protons see Proton Repulsion.

Conclusion

Incremental binding energy may be used to identify the nature (attraction or repulsion) of the nuclear force between nucleons. Second differences in binding energy identify the binding energies due to the interaction of single nucleons. The slopes of the relationships between the incremental binding energy of neutrons and the number of the neutrons and number of protons establish that the interaction between a neutron and proton is an attraction and that the interaction between two neutrons is a repulsion. All of the relationships that can be derived from the binding energies of 2931 nuclides reveal this fact.

The binding energies resulting from the formation of spin pairs are an order of magnitude greater than those due to interactions through the so-called nuclear strong force. The structures of nuclei are largely determined by spin pair formation. Such formations are exclusive in the sense that one neutron can pair with one other neutron and one proton. This leads to chains of nucleon composed of sequences of the form -n-p-p-n- or equivalently -p-n-n-p-. These are called alpha modules. These chains of alpha modules close to form rings. These are what hold nuclei together.

The interactions of nucleons can be explained in terms of their having nucleonic charges. The force between nucleons is proportional to the product of their nucleonic charges. The nucleonic charge of a neutron is smaller in magnitude and opposite in sign to that of a proton. This accounts for unlike nucleons being attracted to each other and like ones repelled.

The model leads to a statistical regression equation that explains 99.995 percent of the variation in the binding energies of 2931 nuclides.

There is much left to be done concerning this matter, but the evidence is clear that while the strong force between protons and neutrons is an attraction it is a repulsion between neutrons. This should not be too much of a surprise; it is just another case of like particles repelling each other.

For the full story of what holds a nucleus together see NUCLEUS.


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