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The Forces Between the Substructures of Nuclei:
Neutron and Proton Spin Pairs, a Singleton Neutron
or Singleton Proton and Alpha Modules

It is often said that nuclei are composed of neutrons and protons (collectively designated as nucleons) but that is like saying nuclei are composed of quarks. Both statements are true but not relevant for analyzing the structure of nuclei. There is no question but that neutrons are combined where possible into spin pairs and the same goes for protons. Any neutron not in a spin pair is called a singleton neutron and likewise for a proton. But there cannot be both a singleton neutron and a singleton proton because they would join into a neutron-proton spin pair and no longer be singletons.

The binding energy of a deuteron (a nucleus consisting of one proton and one neutron) is on the order of two million electron volts (MeV). Those of a triteron (one proton and two neutrons) and a He3 nucleus (two protons and one neutron) are on the order of eight MeV. But an alpha particle with two protons and two neutrons has a binding energy on the order of 28 MeV. The binding energies of the larger nuclei are of an amount that could largely be accounted for by the nucleons where possible being organized into alpha particles. Alpha particles arise in nuclear fission and in spontaneous emission suggesting that alpha particles exist as such within nuclei. But the alpha particle substructure theory of nuclear structure is not completely satisfactory so the theory continues to exist neither confirmed nor denied.

An alternate theory accounts for the same things that the alpha particle theory accounts. Spin pairing is exclusive in the sense that one neutron can pair with one other neutron and with one proton. Thus there can be chains involving modules of the form -n-p-p-n- (or, equivalently -p-n-n-p-). These chains can close and thus form shells as illustrated schematically below.

The first shell would be simply an alpha particle. This corresponds to the magic number of 2 for neutrons and protons.

An alpha module would account for the same binding energy as an alpha particle.

The data on binding energy can be reorganized from a dependence on the number of neutrons and protons into a form depending on the number alpha modules, the number of neutron and proton spin pairs not in alpha modules and a singleton neutron or proton. This data can then be used to compute the incremental binding energies for the various substructures. The incremental binding energies can then be used to estimate the binding energy accounted for by the interactions of the various substructures. This information will tell if there is an attraction or repulsion between each pair of substructures. A positive value indicates an attraction and a negative value a repulsion.

Strong Force Charge

According to previous analysis, if the strong force charge of a proton is designated as +1 then the strong force charge of a neutron is −2/3. The force between two particles is proportional to the product of their charges. Thus like particles repel each other and unlike particles attract each other.

An alpha module, being made up of two protons and two neutrons, has a strong force charge of +2/3. A proton pair has a strong force charge of +2 and a neutron pair a charge of −4/3. Thus a neutron pair should be attracted to an alpha module and repelled by a proton pair. Likewise a proton pair and an alpha module should repel each other.

According to the conventional theory all nucleons attract each other, at least in close proximity. Thus, according to conventional theory, there should be no evidence of nucleonic substructures repelling each other.

The Incremental Binding Energies of Neutron Pairs

To illustrate the advantage of looking at nuclei in terms of the substructures of which they are composed consider the matter of the incremental binding energies of neutrons. Take the case of the isotopes of Tin (proton number 50). The graphs is shown below.

The sawtooth pattern is evidence of the formation of neutron pairs. The sharp drop at 82 neutrons is the effect of the filling of a neutron shell. Now instead of considering the change in binding energy for an increase of one in the number of neutrons, let us look at the change for an increase in the number of neutrons of two; i.e., for one neutron pair. The result is as follows.

For the shell that corresponds to the neutron and proton shells for 51 through 82 particles this is generally linear relationship. There is a slight upward curvature. A quadratic regressiion function explains 99.9 percent of the variation in the incremental binding energy. Effectively the thirty one data points can be reduced to the three parameters of the quadratic regression function. The slope of the relationship is the binding energy due to the interaction of the last neutron pair with the next-to-last neutron. The value is −0.6 MeV, indicating that the force between neutron pairs is a repulsion.

The above relationship was for the isotopes of Tin. The number of protons for this case is 50, which is 25 proton pairs. When the incremental binding energies are computed for the case of 26 proton pairs and the values for 25 proton pairs subtracted the result is the following.

The results are positive indicating that the force between a proton pair and a neutron pair is an attraction.

The Incremental Binding Energies of Proton Pairs

The relationship for proton pairs is much the same as for neutron pairs. The graph for the incremental binding energies of proton pairs is similar but not identical to that neutron pairs.

The downward slope indicates that the force between proton pairs is a repulsion. The magnitude of that slope is −1.4 MeV per proton pair. A linear regression equation explains 99 percent of the variation in the incremental binding energy of proton pairs.

It was previously found that the binding energy of the interaction between two neutrons is −0.6 MeV. The ratio of the binding energy for neutrons to that of protons should be equal to the square of the strong force charge of a neutron to that of a proton. The square root of (0.6/1.4)=0.4286 is 0.655, very close to the 2/3 being used in the analysis.

The Incremental Binding Energies of Alpha Modules

The data can be sorted to give the binding energies of nuclides with increasing numbers of alpha modules but with the numbers of the rest of the substructures held constant. The increments in the binding energies for nuclides with one proton pair, one neutron-proton pair and none of the other substructures is as follows.

The range of the number of alpha modules from 1 to 22 covers several neutron and proton shells. Generally the slope of the relationship is negative thus indicating that the alpha modules repel each other. However for the shell that begins with 15 alpha modules and corresponds to neutrons and protons being in the 29 to 50 shell the level is roughly constant, indicating that the alpha modules are neither attracted or repelled from each other.

The Binding Energy Due to the Interaction
of Alpha Modules and Proton Pairs

When the incremental binding energies are computed for nuclides with one more proton pair than in the above case and the results of the above case subtracted from them the result is;

The values are generally negative. The positive values occur at the points where there is a transition from one shell to another and should be ignored. Thus the results indicate that proton pairs are repelled by alpha modules.

The Binding Energy Due to the Interaction
of Alpha Modules and Neutron Pairs

The incremental binding energies of alpha modules in nuclides with one neutron pair and one neutron-proton pair is quite similar in shape to the one above for the nuclides with one proton pair and one neutron-proton pair. They are virtually but not quite precisely identical.

When the incremental binding energies of alpha modules are computed for nuclides with two neutron pairs and the values for the case of one neutron pair subtracted from them the values are all positive.

Thus the force between a neutron pair and an alpha module is an attraction.

The Binding Energy Due to the Interaction of Alpha Modules
and a Singleton Neutron-Proton Pair

A singleton neutron-proton pair is one in which the neutron is not paired with another neutron and the proton is not paired with another proton. The incremental binding energies of the alpha modules were computed for the nuclides without such a singleton neutron-proton pair (or any other substructure) and with such a pair. The values are shown in the table below along with their differences.

Number of
alpha modules
IBE of α
w/o np pair
IBE of α
w np pair
Difference
in IBE of α
1 28.203836 29.770027 1.566191
2 35.662218 32.7561 -2.906118
3 35.457608 39.907925 4.450317
4 33.025523 32.710575 -0.314948
5 37.612031 36.7761 -0.835931
6 38.28 37.74884 -0.53116
7 35.24377 38.71076 3.46699
8 34.93504 34.96063 0.02559
9 35.3363 35.08177 -0.25453
10 33.4227 34.0398 0.6171
11 35.9873 35.6736 -0.3137
12 36.235 36.2681 0.0331
13 36.291 36.1044 -0.1866
14 31.004 34.3748 3.3708
15 30.958 31.062 0.104
16 30.45 30.54 0.09
17 30.7 30.49 -0.21
18 31 30.9 -0.1
19 31.7 31.6 -0.1
20 31.1 31 -0.1
21 30.5 31 0.5
22 30.7 30.7 0
23 31.3 30.8 -0.5
24 31.5 31.3 -0.2
m

The large positive values occur 1, 3, 7 and 14 alpha modules. These numbers correspond to 2, 6, 14 and 28 neutrons and protons, which are magic numbers corresponding to transitions from one filled shell to the next. The other significant positive values occur at 10 and 21 alpha modules that correspond to neutron and proton numbers of 20 and 42. These correspond to filled subshells. So other than the transitions between filled shells or subshells the values are negative, thus indicating that the force between alpha modules and a neutron-proton pair is a repulsion.

The Binding Energy Due to the Interaction of
Alpha Modules and a Singleton Neutron

The binding energies for nuclides without and with a singleton neutron were compiled and the incremental binding energies of the alpha modules computed. The differences are shown below.

Generally the values are positive indicating that the force between an alpha module and a singleton neutron is an attraction. The significant negative values occur for numbers of alpha modules corresponding to filled neutron and proton shells, the magic numbers.

The Binding Energy Due to the Interaction of Alpha Modules and a Singleton Proton

The differences in the incremental binding energies of alpha modules in nuclides without and with a singleton proton are generally negative, as shown below. This indicates that the force between alpha modules and a singleton proton is a repulsion.

In this case the anomalous positive values cannot be associated with the transitions between shells.

The Binding Energy Due to the Interactions
of Neutron Pairs and a Singleton Neutron or Proton

For this case the differences are unambiguously negative, as shown below.

Thus definitely the force between a neutron pair and a singleton neutron is repulsion.

On the other hand the differences for a singleton proton are unambiguously positive.

Thus the force between a neutron pair and a singleton proton is attraction.

The Binding Energy Due to the Interactions
of Proton Pairs and a Singleton Neutron or Proton

For the interactions of proton pairs with a singleton neutron the results, as shown below, are all positive.

Thus there is an attraction between a proton pair and a singleton neutron.

The interactions between proton pairs and a singleton proton are all negative.

Thus the force between a proton pair and a singleton proton is a repulsion.

Conclusion

The theory that protons and neutrons bear strong force charges of opposite sign is borne out for the cases considered. Alpha modules are attracted to neutron pairs and repelled from proton pairs. Neutron pairs are repelled by each other, as are proton pairs. Alpha modules are, on balance, repelled by each other but perhaps not for those within a shell. Alpha modules and proton pairs are attracted to singleton neutrons and repelled by singleton neutrons. Neutron pairs are attracted to singleton proton and repelled by a singleton neutron.

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