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 separations can be stronger than the electrostatic repulsion between protons but at larger separations can be weaker. This hypothetical force is called the nuclear strong force. There is no more evidence for its existence than that nuclei containing multiple 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. However, spin pair formation is exclusive in the sense that one proton can form a spin pair with one other proton and with one neutron and no more. 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. Spin pairing is more in the nature of a fixed separation linkage.

There is however a force field involving the nonexclusive interaction of nucleons. It will be called the force of nucleonic interaction.

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.

• There are no direct measurements of the forces between nucleons but their natures could be deduced from the changes in potential energy which result from changes in the structure of nuclei. If the force between two objects is negative then there is loss of potential energy as a result of that force when two objects come closer together. On the other hand if the force is positive (a repulsion) there is a gain in potential energy as a result of a reduction of the separation distance when the two objects come closer together. See Force for more on this topic.
• Measured binding energy serves as a measure of the potential energy lost in the formation of nuclei. It is the amount of energy that must be supplied to break up a nucleus into its constituent nucleons. See Binding for more on this topic.

  The binding energy of a nuclide is expressed in millions of electron volts (MeV). This is the amount of energy an electron acquires through falling through an electrical potential difference of one million volts.

• The incremental binding energy for a proton in a particular nuclide is the difference between the binding energy of that nuclide and that of a nuclide that has one less proton.The incremental binding energies (IBE) of nucleons show the effect of the addition of a nucleon. Here is an example.
  

  The odd-even fluctuation in the IBE of a proton is evidence of the formation of proton-proton spin pairs. This is only the data for nuclides with 36 neutrons, but all of the data for the nearly three thousand nuclides show the same phenomenon. The evidence for the formation of proton-neutron spin pairs is the sharp drop in the incremental binding of a proton when the number of protons exceeds the number of neutrons. This is shown above at 36 protons.

  Likewise the IBE for neutrons show the formation of neutron-neutron and proton-neutron spin pairs.

• The data on IBE show that protons and neutrons are organized in nuclear shells. When a shell is filled the IBE for the next nucleon drops off sharply. This is shown in the above graph at 28 protons.

  The conventional values for filled nucleon shells are 2, 8, 20, 28, 50, 82 and 126. The IBE data indicate that 6 and 14 may be filled-shell numbers with 8 and 20 representing filled subshells. A simple algorithm explains the generation of the sequence 2, 6, 14, 28, 50, 82 and 126. Evidence for the existence of nucleonic subshells is given in Subshells. "

• Analysis shows that the second differences in binding energies (the increments in the increments) give the magnitudes of the various interactions between nucleons. See Second Differences for that analysis.

  To eliminate the distracting influence of the odd-even fluctuation due to spin pair formations the data can be given in terms of nucleon pairs.

  The cross differences are roughly constant over the range of a nucleon shell. Therefore the slope of the relationship between the IBE for a proton pair and the number of neutron pairs in the nuclide gives the interaction binding energy between the last proton pair and the last neutron pair added to the nuclide. Here is that relationship for nuclides with 24 proton pairs.

  

  The upward slope to the right indicates that the interaction force between a proton pair and a neutron pair is an attraction. The near linearity of the relation indicates that the interactions of all the proton pairs in a shell with a neutron pair are almost all the same.

• The second differences in binding energy with respect to the number of proton pairs give the interactive binding energies between the last two nucleon pairs added to the nuclide. This slope is found to be positive for all cases and thus the force between any proton pair and any neutron pair due to the strong force is an attraction.

  On the other hand the slope of the relationship between the IBE for a proton pair and the number of proton pairs in the nuclide gives the binding energy due to the interaction of the last two proton pairs to be added to the nuclide. This slope is found to be negative indicating that the force between two protons is a repulsion. Here is an example.

  

  The downward slope to the right of the relation indicates that the reactive force between two proton pairs is a repulsion from the nucleonic force as well as the electrostatic force. The near linearity beyond the 15th proton pair indicates the constancy of the interaction for all proton pairs in the shell.

  This is just one example, but there is exhaustive demonstration that this true in all cases.

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

• The nucleons (neutrons and protons) of a nucleus, whenever possible, form spin pairs (proton-proton, neutron-neutron and proton-neutron). As noted previously, such spin pair formation is exclusive in the sense that one nucleon can form a spin pair with one other nucleon of the same type and with one nucleon of the other type. A nucleon can have interaction with any number of other nucleons through a nucleonic force. Conventionally this was the so-called nuclear strong force . The term strong is inappropriate because that force is not all that strong compared to the forces involved in spin pairing. A more appropriate name for the so-called strong force is the nucleonic force, the force between nucleons.

  A proton spin pair and a neutron spin pair can form an alpha particle whose binding energy might be significantly different from the sum of the binding energies due to the spin pairs within it. More generally the nucleons are linked together in chains containing sequences of the form -n-p-p-n-, or equivalently -p-n-n-p-, which will be called alpha modules. The chains of alpha modules form rings in shells. The lowest shell is just an alpha particle.

• The data for the nuclides were tabulated in terms of their numbers of alpha modules and the number of spin pairs outside of the alpha modules. Nuclides may also contain a singleton (unpaired) proton or neutron. These do not involve any contribution to binding energy due to pairing and they are left out of the analysis for now but will be reconsidered later.

  The binding energy of a nuclide is also affected by the interaction through the nucleonic force of the nucleons. If n and p are the numbers of neutrons and protons, respectively, the number of interactions of the three types are ½n(n-1), np and ½p(p-1).

  The Regressions

  The regression of the binding energies of the 2931 nuclides on the numbers of alpha modules and other spin pairs and on the numbers of nucleonic force interactions gives the following.

  Regression Results
  Variable	Coefficient (MeV)	t-Ratio
  α	42.07905	764.2
  nn	13.89456	151.9
  pp	14.50004	44.6
  np	12.62388	44.1
  p(p-1)/2	-0.54606	-87.1
  np	0.29681	73.5
  n(n-1)/2	-0.20415	-75.6
  Const.	-44.40578	-85.9

  The coefficient of determination (R²) for this equation is 0.99982 and the standard error of the estimate is 6.7 MeV. The average of the binding energies is 1071.9 MeV so the coefficient of variation for the erros of the regression equation is 0.625 of 1 percent.

  The t-ratio for a coefficient is the ratio of its value to its standard deviation. The magnitude of the t-ratio must be two or greater for the coefficient to be statistically significantly different from zero at the 95 percent level of confidence. As can be seen the values of the t-ratios indicate that the likelihood that their values are due solely to chance is infinitesimally small.

  All of the coefficients for the spin pairs are positive indicating the associated force is attractive. They are also approximately of the same magnitude, roughly 14 MeV.

  The coefficients for the interaction of nucleons through the so-called strong force are especially interesting. The coefficients for the interactions of like nucleons are both negative indicating that the forces between like nucleons are repulsions The coefficient for the interaction of unlike nucleons is positive, indicating that the force between unlike nucleons is an attraction.

• The force between like nucleons being a repulsion and being an attraction between unlike nucleons is explained by neutrons and protons having nucleonic charges. The force between two nucleons is proportional to the product of their nucleonic charges. The charges of the proton and neutron differ in sign. Thus if two nucleons are alike the product of their charges is positive and the force is a repulsion; if they are unlike the sign of the product is negative and the force is an attraction.
• If the nucleonic charge of a proton is taken to be 1 and that of a neutron is denoted as q then the interaction between two neutrons is proportional to q² whereas that between a proton and a neutron is proportional to q. Thus the ratio of the interaction between protons to the interaction of a proton and neutron should be equal to q. The estimates of the interactions in the above table give

  q = −0.20415/0.29681 = −0.678658274

  Since q is most likely the ratio of small integers this means that q is equal to −2/3.

• The interaction between protons is complicated by the effect of the electrostatic repulsion. The interaction between protons is proportional to (1+d) rather than 1. Thus the ratio of the interaction between a proton and a neutron and the interaction between proton is proportional to q/(1+d). From the above table that ratio is −0.54354. The value of d has been estimated in another study to be about 0.2. Thus the value of q is equal to −0.54354*(1.2), which is −0.65, again essentially −2/3.

  If the ratio of the interaction of neutrons to the interaction of proton, which should be proportional to q²/(1+d), is used the estimate of |q| is 0.664.

• The statistical performance of the regression equation can be improved slightly .by including any singleton proton and singleton neutron (no more than one of each). The improvement is from a coefficient of determination of 0.99982 to 0.99990. It is significantly improved also by taking into account the shells that the nucleon structures are in. The coefficient of determination for a two way classification of shell occupations is 09999434. For a three-way classification it is 0.99995.

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.

What follows is a graphical analysis. A strictly algebraic analysis is available at Binding Energy Equation.

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 establish that the interaction between two protons is a repulsion. The slopes of the relationships between the incremental binding energy of protons and the number of neutrons establish that the interaction between a proton and a neutron is an attraction.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. 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 larger 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.

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