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The Source of Binding Energy in Nuclides

The binding energy of a nuclide is its mass deficit expressed in energy units via the Einstein formula E=mc². The mass deficit of a nuclide is the difference between its mass and the sum of the masses of its presumed constituents, usually protons and neutrons. The mass deficits of nuclei are an enigma but nevertheless their regularities can be tabulated. In particular one can establish how much the binding energy changes when a particle or a combination of particles is added to it. For example, consider all of the nuclides which could contain an integral number of alpha particles, hereafter called alpha nuclides. Then consider all nuclides that could contain an integral number of alpha particles plus one extra neutron. The differences indicate the effect of an additional neutron on the binding energy of nuclides. The graph of the increments in binding energy due to an additional neutron are shown below.

In contrast consider the corresponding graph for the effect of an additional proton.

At a maximum a proton increases the binding energy by about 3 MeV but generally its effect is much smaller and for larger alpha nuclides it is negative. In contrast the effect of the additional neutron seems to be asymptotically approaching a level of about 14 MeV. A graph of the two effects gives a more striking comparison.

As a tentative conclusion one could say that protons contribute very little to the binding energies of nuclides. To work towards a corroboration of this tentative conclusion consider the effect of adding a proton-neutron pair to the alpha nuclides.

After the usual small nuclide irregularities the effect of the proton-neutron pair levels off at about 12.5 MeV. It is worthwhile to compare the effect of the proton-neutron pair with the sum of the effects due to a proton and a neutron separately.

There is some effect for the interaction of the proton and neutron but it is small compared with the effect of the neutron and it is relatively constant.

The effect of a proton-neutron pair can be compared with the effect of a neutron pair. The effect of a neutron pair is shown first.

This has the same general shape as the effect of an additional neutron except the curve is asymptotically approaching a level of about 28 MeV, the binding energy of an alpha particle. This suggests that perhaps the effect of a neutron pair is simply twice the effect of a neutron. The graph of the two quantities shows that this is roughly true but there is an additional effect due to the interaction of the two neutrons just as there was an effect for the interaction of the proton and neutron in a proton-neutron pair.

The increments in the enhancements of binding energy due to an additional neutron show the diminution which is, aside from fluctuations, about inversely proportional to the size of the nuclide.

Conclusion

The dominant component of the binding energy of a nuclide comes for the neutrons it contains. This component is a function of nuclide size with its value asymptotically approaching about 12.5 MeV. There are smaller components which have to do with the protons in the nuclides and the interaction of the protons and neutrons. These are enough smaller that they could be said to be of a lower order of magnitude.

The big puzzle in nuclear structure is the relatively large value of binding energy for the alpha particle, the helium 4 nuclide. From what was presented above the binding energy of the alpha particle is about what would be expected of a nuclide with two neutrons and two protons. The real puzzle is the very low binding energies for the nuclides smaller than the helium 4 nuclide. That puzzle is dealt with elsewhere.


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