San José State University

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Thayer Watkins
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On the Interaction Binding Energies
of Two Nucleons Being a Function
of the Shells They Occupy

Previous studies established that the second differences in binding energy give the magnitude of the interaction between the last nucleons to be added to a nuclide. Those second differences are equivalent to the slopes of the relationships between the incremental binding energies and the number of nucleons of each type. Those slopes indicate that neutrons and protons are attacked to each other but neutrons repel neutrons and proton repel protons.

The purpose of this material is to introduce the topic that the magnitude of the interaction between two nucleons is a function only of the shells which the two occupy. For example, consider the interaction between the 60th neutron and the various number of protons. The graph of the data is shown below. The interaction binding energies are in units of millions of electron volts (MeV).

There is no apparent trend of interaction energy with the proton number. To confirm this the interaction energy IE was regressed on the proton number P. The results are

IE = 0.42992 + 0.00011P
[2.7]         [0.015]

The values in square brackets, […], are the t-ratios of the coefficients. The t-ratio of a coefficient is the ratio of the coefficient value to the standard deviation of the estimate of the coefficient. For a coefficient to be statistically significantly different from zero at the 95 percent level of confidence its t-ratio must be 2 or larger in magnitude.

The t-ratio of 0.015 indicates the slope of the relationship is not significantly different from zero. Therefore the interactive binding energy between the 60th neutron and a proton is the same over the whole range from 37th to the 56th proton. This range spans two proton shells, the one from 29 to 50 and the one from 51 to 82. There appears to be no difference in the interactive binding energy of the 60th neutron and protons in the two different proton shells.

There is a definite odd-even fluctuation in the pattern. This means that the interaction of a neutron with a proton is different for a proton that is paired with another proton than it is with an unpaired proton. A variable Ev is introduced with Ev=1 if P is even and Ev=0 otherwise. A regression of interactive energy on proton number and the evenness of the proton number the results are:

IE = 1.00877 − 0.00501P + 0.68129Ev
[3.4]     [-0.8]     [9.4]

The t-ratio of −0.8 indicates that the coefficient of P is not statistically signficantly different for from zero at the 95 percent level of confidence.

Since P was not statistically significant it was dropped from the regression and the following results were obtained.

IE = 0.0503 + 0.676[Ev]
[15.4]    [9.5]

The magnitudes are surprising even perplexing. The interaction energy of the 60th neutron is only 0.0503 MeV with an unpaired proton but 0.7263 MeV with a paired proton, for an average ofo 0.3632 MeV. For now the difference in the magnitudes is unexplained.

The interaction energies should be similar in the other direction. Here is the graph of the interaction of the 40th proton with the 41st through 67th neutrons.

There appears to be no apparent trend of interaction energy with proton number but there is a definite odd-even effect. The regression confirms this

IE = 0.18460 −0.000855P + 0.61245Ev
[0.5]     [-0.1]     [6.3]

Then coefficient of P is obviously not statistically significant. Dropping P from the regression results in

IE = 0.13845 + 0.61245Ev
[2.1]     [6.4]

The interaction of the 40th proton with an unpaired neutron is only 0.13845 MeV but is 0.7509 MeV with a paired neutron, for an average of 3755 MeV.

Conclusion

The interaction energies between 60th neutron(which is in the 51 to 82 shell) and protons in the 29 to 50 shell are all the same. Likewise the interaction energies between 40th proton (whichn is in the 29 to 50 shell) and neutrons in the 51 to 82 shell are also all the same.

There is a drastic difference in the magnitude of the interaction energy for a nucleon with an unpaired nucleon of the opposite type compared with a paired one.


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