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 was given the name nuclear strong force. There is no more evidence for its existence than that there exists nuclei containing multiple protons which 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 pairing involves pairing together particles of opposite spin. It is much like placing together two bar magnets with opposite poles aligned.

Spin pair formation is exclusive in the sense that one neutron can form a spin pair only 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. Spin pairing is more in the nature of a linkage with a fixed separation distance between particles than a force involving a force field with distance dependence. There is however a force field involving the nonexclusive interaction of nucleons. This will be referred to as the force of nucleonic interaction. It is not exclusive.

This is an extremely important topic and therefore the argument will be laid out 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. i>

  The binding energy of a nucleus, in general, is the amount of energy that must be supplied to break it up into its constituant component nucleons. In addition to the binding energy due to the loss of potential energy involved in its formation there is the energy equivalent of its mass deficit. The mass of a nucleus is less than the sum of the masses of the nucleons it is made of.

• Thus true binding energy would include the potential energy lost when the nuclide was formed in addition the binding energy based upon the mass deficit of the nuclide. Only the mass deficit binding energies of nuclides are known. These measured binding energies serve as correlates 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 neutron (IBEn) in a particular nuclide is the difference between the binding energy of that nuclide and that of a nuclide that has one less neutron.This incremental binding energies (IBEn) of neutrons show the effect of the addition of a neutron. Here is an example.
  

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

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

• The data on IBE show that neutrons and protons 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 50 neutrons.

  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 neutron pair and the number of proton pairs in the nuclide gives the interaction binding energy between the last neutron pair and the last proton pair added to the nuclide. Here is that relationship for nuclides with 24 neutron pairs.

  

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

• The second differences in binding energy with respect to the number of neutron 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 neutron pair and any proton pair due to the interaction of their nucleonic charges is an attraction.

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

  The downward slope to the right of the relation indicates that the reactive force between two neutron pairs is a repulsion. The near linearity beyond the 25th neutron pair indicates the constancy of the interaction for all neutron pairs in the shell.

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

Thus neutrons repel each other but are attracted to protons.

What follows is an introduction to the more complete model of nuclear structure.

• The nucleons (protons and neutrons) of a nucleus, whenever possible, form spin pairs (neutron-neutron, proton-proton and neutron-proton). 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 opposite type. A nucleon can also have interaction with any number of other nucleons. Conventionally this is called the 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 would be the nucleonic force, the force between nucleons.

  A neutron spin pair and a proton spin pair can form an alpha particle whose binding energy might be significantly greater than 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) neutron or proton. 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 errors 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 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 protons and neutrons having nucleonic charges. The force between two nucleons is proportional to the product of their nucleonic charges. The charges of the neutron and proton 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 neutron and a proton is proportional to q. Thus the ratio of the interaction between neutrons to the interaction of a neutron and proton 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 neutron and a proton and the interaction between protons 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 protons, 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 neutron and singleton proton (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 0.9999434. For a three-way classification it is 0.9999492.

Force and Second Differences

In general a force F is related to a potential energy function U by the relation

F = −∇U

That is to say, force is equal to the negative of the gradient of the potential energy function. If the force between two objects is a function of their separation distance s, then

F = −(dU/ds)

If F is negative when two objects are motionless then the separation tends to be reduced; i.e., there is an attraction between the objects. If the objects move closer together there is a loss of potential energy. Likewise if F is positive when two objects are motionless then the separation tends to be increased; i.e., there is a repulsion between the objects. If the objects move closer together there is a gain of potential energy.

Binding Energy Equation.

Cross Differences

The increase with respect to the number of protons
in the incremental binding energies of neutrons
is equal to theinteraction 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 nucleonic 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 nucleonic 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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