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through the electrostatic force but also through a nucleonic force, as do neutrons as well. Nuclei are held together by the formation of nucleon spin pairs and the attraction between protons and neutrons. |
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.
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 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 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. "
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.
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.
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 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 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.
Since q is most likely the ratio of small integers this means that q is equal to −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.
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.
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.
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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