San José State University

applet-magic.com
Thayer Watkins
Silicon Valley
& Tornado Alley
USA

Protons repel each other not only through the electrostatic force
but also through through the nuclear strong force, as do neutrons.
Nuclei are held together by the formation of nucleon spin pairs
and the attraction between protons and neutrons.

Background

Conventional nuclear theory holds that there is a force that involves an attraction between all nucleons (protons and neutrons). 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 proton can form a spin pair with one other proton and with one neutron. The same applies to neutrons. 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 protons repel each other but are attracted to neutrons. What follows is an introduction to the more complete model of nuclear structure.



Second Differences

Cross Differences

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

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

   

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

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

   

The subtraction eliminates all the interactions of the n neutrons with each other. It also eliminates the interactions of the p-1 protons with each other and the p-1 protons with the n neutrons.

  

What are left are the interactions of the p-th proton with the other p-1 protons and the interaction of the p-th proton with the n-th neutron. The subtraction eliminates the interactions of the p-th proton with the other (p-1) protons.

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

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

Strict Second Differences

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

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

  

The black squares are to indicate that there is no interaction of a proton 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 proton pair is the difference in the binding energy of the nuclide with p proton pairs and n neutron pairs and that of the nuclide with p-1 proton pairs and n neutron pairs. In the diagrams below the interactions for the nuclide with (p-1) proton pairs and n neutron pairs are colored.

  

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

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

 

Now consider the IBE of proton pairs for p proton pairs and (p-1) proton pairs displayed together. The IBE's for (p-1) proton pairs are displayed in pink.

 

The subtraction of the IBE for (p-1) proton pairs and n neutron pairs from the IBE for p proton pairs and n neutron pairs depends upon the magnitude of the interaction of the (p-1)-th proton pair with the different proton pairs compared to the interaction of the p-th proton pair with those same proton pairs. Visually this is the subtraction the values in the pink squares from the white squares horizontally or vertically. When the p-th and the (p-1)-th proton pairs are in the same shell the magnitudes of the interactions with any neutron pair have been found empirically to be equal. This is from the previous analysis concerning cross differences. Thus the interactions with the n neutron pairs are entirely eliminated.

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

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

For the corresponding analysis for neutrons see Neutron 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 protons and the number of the protons and number of neutrons establish that the interaction between a proton and neutron is an attraction and that the interaction between two protons 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 proton can pair with one other proton and one neutron. 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 proton is smaller in magnitude and opposite in sign to that of a neutron. 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 neutrons and protons is an attraction it is a repulsion between protons. This should not be too much of a surprise; it is just another case of like particles repelling each other.


HOME PAGE OF applet-magic
HOME PAGE OF Thayer Watkins