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Charge of a Neuton to that of a Proton Based on the Limits of Nuclear Stability in Terms of the Number of Neutrons |
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 ssome balance between the number of neutrons and the number of protons for a nuclei 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 protons the minimum number and the maximum number of neutons 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
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 E_{nn} 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 E_{np}. The binding energies associated with the 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.
The energy change associated with the addition of another proton to a nuclide with p protons and n neutrons in which p is odd and less than n is
The minimum number of neutrons for a nuclide with p protons is reached when IBEp≤0. This means that
The binding energy of an additional neutron to a nuclide with p protons and n neutrons in which n is odd and less than p is
For IBEn to be greater than or equal to zero requires a maximum number of neutrons of
Thus the slope of the relationship between n_{max} and p should be 1/ν the same as the slope of the relationship between n_{min} and p. It is therefore reasonable to add together the equations for n_{min} and n_{max} and divide by 2. The result is
The levels of E_{nn} and E_{pp} may vary with n but the value of their difference is not likely to vary as much and the their difference is divided by 2k rather than k. The value of E_{np} is eliminated entirely from the picture.
The graph of the midpoint of the limits, n_{mid}, versus number of protons is shown below.
If one observes closely the previous graph there is change in the slope of the pattern at about 76 protons. Some different phenomena are operating beyond 76. Therefore those points are left out of the analysis for now. The regression equation based on the data from 1 to 76 protons is
The coefficient of determination for the regression is 0.99763 and the t-ratio for the coefficient of p is 176.7. The coefficient of p should be equal to 1/ν; thus ν should be 0.72696, a value close to 0.75. The value of 1.37559 is statistically significantly different from 1/(3/4)=4/3 at the 95 percent level of confidence . The z-ratio is 5.43. And, it is definitely significantly different from the value of 3/2 that would occur for ν=2/3. This is in conflict with the value of 2/3 found when the analysis was based upon maximum and minimum number of protons. Obviously some important factor concerning protons and neutrons has been left out of the analysis.
A bent-liner regression equation can be fitted to the whole range of data allowing for for n_{mid} in terms of p with bend points at 76 and 94 protons. The result is
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 0.99927 and the t-ratio for the coefficient of n is 213.7. The coefficient of p, 1.36976, is the estimate of 1/ν, the reciprocal of the ratio of the magnitude of the nucleonic charge of a neutron to that of a proton. The value of 1.36976 is statistically significantly different from 4/3 at the 95 percent level of confidence. The z-ratio is 5.7. The bend-points of 76 and 94 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 numer of protons for each level of neutrons see Nuclear Charge Ratio .
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