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Estimation of the Nucleonic Charge of
a Neutron Relative to That of a Proton

A previous study verified that the effects on structural binding energy of additional constituents to nuclei such as alpha particles and nucleonic spin pairs can be explained in terms of neutrons and protons having a charge with respect to the strong force. Thus if the nucleonic charges of a neutron and a proton are denoted as sn and sp, respectively, then, for example, that of a deuteron (a neutron-proton pair) is (sn + sp).

The most interesting question is the value of sn/sp. The presumption in conventional nuclear theory is that this ratio is 1.0. This started with Werner Heisenberg in the 1930's after the discovery of the neutron. He conjectured that the neutron and the proton were really just the same particle but in a proton an electrostatic charge was turned on. It was then assumed that all nucleons attracted each other with equal force and it was this strong force between nucleons that holds nuclei together. With the development of Quark Theory in the 1960's it was found that the neutron and proton were not the same, but the presumption of equal attraction among nucleons was kept.

There is evidence (1 and exclusive in the sense that a neutron can pair with one other neutron and with a proton but no more. It is likewise for protons. This means that neutrons and protons can form chains made up of segments 'of the form -n-p-p-n-, or equivalently -p-n-n-p-. These sequencies will be called alpha modules because the simplest such unit is the alpha particle.

Generally a force that is carried by particles the way the electrostatic force is carried by photons and the nuclear strong force is carried by π mesons is determined by a formula of the form

F = HZ1Z2f(s)/s²

where H is a constant, Z1 and Z2 are the charges on the substructures and s is the distance between their centers. For the electrostatic force f(s) is equal to 1 and for the nuclear strong force f(s) is most likely exp(−s/s0). If F is positive then it acts to increase the separation distance. In other words, there a repulsion between the nuclear particles. If F is negative there is an attraction. Clearly for there to be an attraction the signs of the charges must be different. Thus unlike particles attract and like particles repel.

For forces given by the above formula the potential energies are of the form

V(s) = HZ1Z2sf(z)dz

There are mass deficits for nuclei. That is to say, the mass of a nucleus is less than the sum of the masses of its constituent nucleons. This mass deficit expressed in energy terms is called the binding energy of the nuclide. The binding energy is just the loss of potential energy involved in the formation of a nuclide. Some of the binding energy of a nuclide is due to the formation of substructures such as alpha modules and spin pairs and some, called structural binding energy is due to the combination of such substructures in the nuclide.

The effect of adding a constituent such as one neutron can be determined by tabulating the binding energies of all nuclides which could contain only alpha modules then tabulating the binding energies of all that contain only alpha modules plus one neutron. Such effects so measured may be dependent up on which shell the added constituent goes into.

The alpha module shells contain 1, 2, 4, 7, 11, 16, 22 alpha modules. These correspond to shell occupancies for the nuclear magic numbers for neutrons and protons of 2, 6, 14, 28, 50, 82, 126; i.e., 2, 4, 8, 14, 22, 32, 44.

The Effect of an Added Neutron Compared
with the Effect of an Added Proton

The hypothesis is that the effect of an added neutron is always a constant ratio of the effect of an added proton; i.e.,

BEn = σBEp

The scatter diagram for the effects of added singleton neutrons and protons is as follows.

The data points for the first through third shells (marked by red bars in the above diagram) have a pattern entirely different from the remaining data points. They represent the result of moving from one shell to another. The data points for the fourth and fifth shells more or less fit the same pattern. This pattern has a negative slope. The regression of the effect of an added neutron on the effect of an added proton gives the following result.

BEn = 10.66577 −0.68271BEp

The coefficient of determination (R²) is only 0.51410 but the t-ratio for the regression coefficient is −3.7, indicating that the regression coefficient is statistically significantly different from zero at the 95 percent level of confidence.

The value of −0.68271 is quite close the value of roughly −2/3 found in a previous studies.

There should be a similar relationship between the effect of adding a neutron pair and adding a proton pair; i.e.,

SBEnn = σSBEpp

Here the effects of the formation of nucleon pairs, En and Ep, must be taken into account.

BEnn − Enn = σ(BEpp − Epp)
which reduces to
BEnn = (Enn−σEpp) + σBEpp

Again the data points for the first through the third alpha module shells, marked with red bars, do not fit the pattern of the rest of the data points. The pattern for those first though third shell data points is that of the relationship between shells rather than the relationship within shells.

The regression equation using the data points for the fourth and fifth alpha particle shells is

BEnn = 21.47905 −0.76600BEpp

The coefficient of determination (R²) is only 0.30225 and the t-ratio only -2.0 indicating that the regression coefficient is just barely statistically significantly different from zero at the 95 percent level of confidence. But the value is negative and the magnitude essentially the same as the previous estimate of −0.68271.


The evidence here confirms the concept that the effects of additional constituents on the binding energies of nuclides is explained by the neutron and proton having strong force charges. Furthermore these charges are of the opposite signs thus explaining the attraction of neutrons and protons. And, quite significantly, the magnitude of the charge for the neutron is only about two thirds that of a proton.

(To be continued.)

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