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An Estimate of the Ratio of the
Nucleon Charge of a Neutron to
that of a Proton Based on the
Limits of Nuclear Stability

Nuclei are held together by the mutual attraction between neutrons and protons. Neutrons are repelled from each other through the strong force. Protons are also repelled from each other not only through the electrostatic force but also through the strong force. Therefore there has to be some balance between the number of neutrons and the number of protons for a nucleus to hold together. If there are too many protons compared to the number of neutrons the repulsion between the protons overwhelms the attraction between neutrons and protons. Likewise if there are too few protons the repulsion between the neutrons overwhelms the neutron-proton attraction. There is an asymmetry between the numbers of neutrons and protons that indicates the strength of the repulsion between protons due to the strong force is greater than that between neutrons. The strong force drops off faster with distance than the electrostatic force so the electrostatic repulsion between protons becomes relatively stronger in larger nuclides where the average distance between protons becomes greater. The situation is made more complicated by the fact that neutrons form spin pairs with each despite their mutual repulsion and protons do likewise. Spin pair formation is relatively more important for the smaller nuclides.

There are 2931 nuclides stable enough to have had their masses measured and their binding energies computed. For each number of neutrons the minimum number and the maximum number of protons were compiled. The results are displayed in the following graph.

In the graph there is some piecewise linearity displayed.

A Previous study developed evidence that the nucleonic (strong force) charge of a neutron is of the opposite sign and smaller in magnitude from that of a proton. Let ν denote the ratio of the nucleonic charge of a neutron to that of a proton. The actual value of ν is undoubtedly a simple fraction. Previous work indicated that the relative magnitude of the neutron charge could be 2/3 or 3/4. Furthermore such a difference in charge of the nucleons can account for the limits to the values of the proton numbers of the known nuclides, shown above.

Another study demonstrated that the binding energy increments experienced by additional nucleons to a nuclide is a function of two components. One is simply the difference in the number of protons and neutrons in the nuclide. This component has to do with the formation of a neutron-proton spin pair. The other component has to do with the interaction of nucleons through the strong force and it is a function of the net nucleonic charge of the nuclide. If p and n are the numbers of protons and neutrons, respectively, of the nuclide then the net nucleonic charge ζ is

ζ = p − νn

where ν is the magnitude of the nucleonic charge of the neutron relative to that of a proton.

The binding energy associated with the interaction nucleons through the strong force is a nonlinear function of ζ, but for small values of ζ to a reasonable approximation it is kζ, where k is a constant. Nucleons also interact through the formation of spin pairs. For example, the addition of another neutron to a nuclide with an odd number of neutrons would result in the formation of a neutron-neutron spin. Let Enn be the binding energy associated with the formation of a neutron-neutron spin pair. If there are unpaired protons in the nuclide the addition of another neutron would result in the formation of a neutron-proton spin pair with a binding energy of Enp. The binding energies associated with the formation of spin pairs are not really constants independent of the levels of n and p but for the present they are assumed to be constants.

An Additional Neutron

The energy change associated with the addition of another neutron to a nuclide with p protons and n neutrons in which n is odd and less than p is

IBEn = kζ + Enn + Enp

The minimum number of protons for a nuclide with p protons is reached when IBEn≤0. This means that

k(pmin − νn) + Enn + Enp = 0
and hence
pmin = νn + Enn/k + Enp/k

An Additional Proton

The binding energy of an additional proton to a nuclide with p protons and n neutrons in which p is odd and less than n is

IBEp = −kζ + Epp + Enp
and thus
IBEp = kνn - kp + Epp + Enp

For IBEp to be positive requires a maximum p of

pmax = νn − Epp/k − Enp/k

Thus the slope of the relationship between pmax and n should be ν the same as the slope of the relationship between pmin and n. It is therefore reasonable to add together the equations for pmin and pmax and divide by 2. The result is

pmid = (pmin + pmax)/2 = νn + (Enn−Epp)/(2k)

The levels of Enn and Epp may vary with n but the value of their difference is not likely to vary as much and their difference is divided by 2k rather than k. The value of Enp is eliminated entirely from the picture.

The graph of the midpoint of the limits, pmid, versus number of neutrons is shown below.

If one observes closely the previous graph there is change in the pattern at about 120 neutrons. Some different phenomena are operating beyond 120. Therefore those points are left out of the analysis for now. The regression equation based on the data from 1 to 120 is

pmid = 0.67264n + 4.30546

The coefficient of determination for the regression is 0.99735 and the t-ratio for the coefficient of n is 210.7. The value of 0.67264 is not statistically significantly different from 2/3 at the 95 percent level of confidence even though the standard deviation of the estimate is only 0.0031917. The z-ratio is 1.87. This is a resounding confirmation of 2/3 being the ratio of the magnitude of the nucleonic charge of the neutron to that of the proton. The actual ratio is of course −2/3.

A bent-liner regression equation can be fitted to the whole range of data allowing for pmid in terms of n with bend points at 120 and 142 neutrons. The result is

pmid = 0.66849n −0.35247u(n-120) + 0.51886u(n-142) + 4.47280

where the function u(z) is equal to z if z≥0 and 0 if z<0.

The coefficient of determination for the regression is 0.99854 and the t-ratio for the coefficient of n is 234.5. The coefficient of n, 0.66849, is the estimate of ν, the ratio of the magnitude of the nucleonic charge of a neutron to that of a proton. The value of 0.66849 is not statistically significantly different from 2/3 at the 95 percent level of confidence. The z-ratio is 0.64. The bend-points of 120 and 142 are not necessarily the ones that maximize the coefficient of determination but they are close.

For the corresponding analysis in terms of the maximum and minimum number of neutrons for each level of protons see Nuclear Charge Ratio 2.


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