San José State University

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 The Binding Energies of Small Nuclides

One of the major enigmas of the physics of nuclei is the relatively low values of the binding energies of the small nuclides compared to that of the alpha particle. one approach to explaining the binding energies is in terms of the number of strong force interactions of the neutrons and protons in a nuclide and the number the various types of spin pairs that can be formed. The deuteron has only the neutron-proton strong force interaction and the formation of the one neutron-proton spin pair. The triteron has three strong force interactions; two neutron-proton interactions and one neutron-neutron interaction. It has one possible neutron-neutron spin pair and one neutron-proton spin pair. The alpha partice, on the other hand, has four neutron-proton strong force interactions, one neutron-neutron interaction and one proton-proton interaction. There can be four possible spin pairs formed; one neutron-neutron pair, one proton-proton pair and two neutron-proton pairs. These differences could possibly explain a deuteron having a binding energy of 2.22457 million electron volts (MeV), a triteron 8.48182 MeV and an alpha particle having a binding energy of 28.29 MeV. It would be very easy to choose values for the binding energies associated with the various interactions and spin formation that would account for the binding energy values, but those values would not necessarily be consistent with the binding energies of the other nearly three thousand nuclides.

Let the numbers of neutrons and protons be denoted by #n and #p, respectively. The number of possible neutron-neutron interactions is then #n(#n-1)/2 and this number will be denoted as #nn. Likewise the number of possible proton-proton interactions is #p(#p-1)/2 and this is denoted as #pp. The number of neutron-proton interactions is #n#p. The number of neutron-neutron spin pairs is #n%2; i.e., the largest integral number in the ratio #n/2 and likewise #p%2 for the proton pairs. The number of possible neutron-proton spin pairs is min(#n, #p).

When the binding energies of all nuclides except the single neutron and single proton are regressed on the above six variables the result is

 variable min(n,p) #n%2 #p%2 #np #nn #pp coefficient 10.31174 6.91730 2.15500 0.27804 -0.19254 -0.48958 t-ratio 34.7 76.9 8.3 35.9 -37.3 -40.8

The coefficient of determination (R²) reported by EXCEL for this regression is 0.99988, but this exaggerates the performance because this regression has no constant term so the unexplained variance is being compared to the average squared value of the dependent variable rather than its variance (average square deviation from the mean). The corrected R² value is 0.99935, still a very high value. The standard error of the estimate for the regression is 12.88 MeV, which is only 1.2 percent of the average binding energy. However this standard error suggest that the equation would not give an accurate estimate of the binding energy of the deuteron which is only 2.22457 MeV; and it does not. The equation prediction for the binding energy of the deuteron is 10.6 MeV. However for the alpha particle the equation estimate is 29.57 MeV, which is not bad as an estimate of 28.29 MeV. On the other hand, the equation estimate for the triteron is 17.59 MeV when the actual value is 8.48182 MeV.

While the regression equation does not give accurate estimates of the binding energies of the nuclides smaller than the alpha particle there are several very important implications of the results.

• The strong force interactions of neutron and neutrons is negative which indicates a repulsion. Likewise the proton proton interaction is negative and there is a strong force repulsion of protons for each other as well as for the electrostatic force. The strong force interaction between neutrons and protons is positive and thus there is a strong force attraction between unlike nucleons.
• The magnitudes of the strong force interactions differ. The values can be explained, at least approximately, in terms of nucleonic charges. The value for the neutron-neutron interaction is the negative of the square of the nucleonic charge for a neutron. Thus the strong force charge of the neutron is (0.192544389)½=0.438798803. Likewise the value for the strong force charge of the proton is (0.489577522)½=0.699698165. Therefore the neutron-proton interaction should be 0.438798803*0.699698165=0.307026717. The actual value is 0.278037752, which is only about 10 percent less. Another way of expressing the relationship is that the neutron-proton interaction is approximately equal to the geometric mean of the neutron-neutron interaction and the proton-proton interaction.
• For the small nuclides the overwhelming proportion of the binding energy comes from the formation of nucleonic pairs. The strong force element of the binding energy becomes comparable in magnitude to the pair formation component at something on the order of sixty nucleons.
• The binding energies due to the three type of pairs differ considerably, with neutron-proton pairs being the strongest and proton-proton pairs the weakest.

(To be continued.)