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The Binding Energy of Protons
Which is Due to the Interaction of a
Proton with the other Nucleons
Through the Nuclear Strong Force

Let BE(n, p) denote the binding energy of a nuclide with n neutrons and p protons. The incremental binding energy (IBEP) of the p-th proton in a nuclide with n neutrons and p protons is then

IBEP(n, p) = BE(n, p) − BE(n, p-1)

The IBEP represents the effects of an additional proton on binding energy due the possible formation of spin pairs and the interaction of that additional proton with the other protons and the neutrons of the nuclide. A proton-proton spin will be formed if and only if there is an unpaired proton in the nuclide with n neutrons and (p-1) protons; i.e., if (p-1) is an odd number. A neutron-proton spin pair will be formed if p is less than n.

A previous study estimated the binding energies due to the formations of neutron-neutron and proton-proton spin pairs. That study found that the energies due to the formation of those types of spin pairs have distributions which are approximately normal (Gaussian). The mean value for the formation of a neutron-neutron spin pair is 1.956 million electron volts (MeV). For the formation of a proton-proton spin pair the mean value is 2.856 MeV.

In the process of obtaining those estimates of the binding energies due to the formation of spin pairs the values of binding energy in the incremental binding energies of neutrons and of protons which are due to the interaction of the additional neutron or proton with the other nucleons through the strong force were computed. It was found that the incremental binding energy of a neutron through the strong force interactions (IBEPSF) is approximately a linear function of the number of protons in the nuclide. If that linear function is projected backwards to 0 protons that is the effect of the neutrons on binding energy. That value is simply the value of the constant in the linear relation. The ratio of that constant to the number of neutrons is the amount of binding energy due to the strong force interaction of one proton and one neutron. The slope of the linear function is the amount of binding energy due to the strong force interaction of two protons.

In order to eliminate the complicating factor of the formation of neutron-proton spin pairs the analysis was limited to values of p which are greater than n. This severely limited the data points for the analysis.

Regression analysis can be used to obtain values for the constant and the slope of the linear function for the isotopes of an element. It is presumed that the function explaining the binding energies due to strong force interaction is of the form

IBEPSF = c0n + c1p

Regression analysis applied to the 57 observations for IBEPSF gives

IBEPSF = 0.6278n − 0.5226p
[7.9] [−7.9]

The numbers in square brackets [ ] are the t-ratios for the coefficients. The t-ratio for a coefficient is the ratio of the value of the coefficient to its standard deviation. For a coefficient to be statistically significantly different from zero at the 95 pecent level of confidence its magnitude must be greater than or equal to 2. The coefficients in the above equation are highly significant. The coefficient of determination (R²) for this equation is 0.532 and the standard error of the estimate is 0.8 MeV.

The negativity of the coefficient for the interaction of two protons indicates that the strong force between two protons is repulsion. The positivity of the coefficient for the interaction of a proton and a neutron indicates that the strong force between a proton and a neutron is attraction.

In order to see how much the coefficient of determination could be increased by utilizing quadratic functions two addition regressions were run.

IBEPSF = 1.0625n −0.7132p −0.009166n*p
[12.4] [-12.8] [-6.9]

The coefficient of determination (R²) for this equation is 0.752 and the standard error of the estimate is 0.6 MeV. These are significant improvements in the statistical performance of the regression equation or the previous one. .

IBEPSF = 1.6386n −1.1846p + 0.1335n*p −0.06337p² −0.05735n²
[6.5] [-5.8] [1.4] [-1.5 [-1.1]

The coefficient of determination (R²) for this equation is 0.780 and the standard error of the estimate is 0.57 MeV. Not much was gained by adding the two quadratic terms to the equation. in terms of the coefficient of determination but the reduction in the standard error of the estimate is notable. The signs and relative magnitudes of the coefficients for proton and neutron interactions are maintained in the three regressions. However the coefficients of the quadratic terms are not significantly different from zero at the 95 percent level of confidence.

A Comparison

The binding energy due to the strong force interaction of a proton and a neutron was found, in the first regression in this study, to be 0.6278 MeV., A previous study based upon the incremental binding energy of neutrons found 0.4532 MeV. These values are different but, given the standard deviations of both estimates, are the significantly different statistically. The standard deviations of the two estimates are 0.006364 and 0.07949 MeV. The standard deviation of the difference is the square root of the sum of the squares of these two numbers. That value is 0.07974 MeV. The difference in the estimates is 0.1746 MeV. The ratio of this difference to its standard deviation is 2.19 so the difference is just barely significant at the 95 percent level of confidence.

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