Almost all nuclides have a mass deficit; i.e., their masses are less than the sum of the masses of the neutrons and protons of which they are made up. The energy equivalent of the mass deficit of a nuclide is called its binding energy. That binding energy represents the energy required to break up a nuclide into its constituent nucleons (neutrons and protons).

The purpose of this webpage is to show that the binding energy of a nuclide is the same as the loss of potential energy involved in its formation.

Potential Energy

When a force exists between two particles the work that has to be exerted in order to increase their separation is called the increase in potential energy of the system of the two particles. The work is the negative of the product of the force and the increase in the separation distance. The work is carried out slowly so as not to generate any kinetic energy for the particles. If the particles are allowed to return to their previous state, according to the principle of the conservation of energy, the work done in separating them will be returned in some form such as kinetic energy or the energy of an emitted photon.

When a neutron and proton come together to form a deuteron there is a gamma ray of energy of about 2.2 million electron volts (MeV) emitted. The deuteron has a mass deficit, but when the deuteron is broken up, such as when it absorbs a gamma ray of at least 2.2 MeV in energy the missing masses of the two nucleons is regained.

There is a force between a neutron and a proton. If it were possible to separate the neutron and proton in a deuteron by a small amount the energy that would have to be supplied would be that of the recovery of mass for the nucleons. Thus the work in separating the nucleons of a composite nucleus is just the binding energy of the nucleus.

The Formation of Nucleonic Spin Pairs

There are three type of pairs that can be formed from neutrons and protons; neutron-neutron, proton-proton and neutron-proton. Each of these shows up in terms of binding energy. This is most easily seen in terms of the incremental binding energies. For example, for neutron-neutron and proton-proton pairs this shows up as the sawtooth pattern in the displays below.

For neutron-proton pairs the effect on binding energy shows up as a sharp drop in incremental binding energy when the number of neutrons exceeds the number of protons in a nuclide or vice versa.

The Effect of the Electrostatic Repulsion
between Protons on Binding Energy

The binding energy of a triteron (one proton and two neutrons) is 8.48 MeV whereas that of He3 (two protons and one neutron) is 7.72 MeV. The difference of 0.76 MeV is largely due to the electrostatic repulsion between the protons in the He3 nucleus that doesn't exist in the triteron. Some of the difference may be due to a difference in the binding energy of a proton-proton pair and a neutron-neutron pair.

The effect of the electrostatic repulsion on binding energy is very regular and predictable. In the following displays alpha nuclide refers to nuclides that could be composed entirely of alpha particles; e.g., Be8.

This indicates that binding energy indicates not only the potential energy associated with the nuclear strong force but also any other force involved in nuclei.

Quantization and Photon Emission

For the electrons in an atom when an electron changes state a photon is emitted. This is because the potential energy and the kinetic energy of the electron are quantized and the allowable change in potential energy is not compatible with the allowable changes in kinetic energy so the energy difference goes into the emission of a photon. The fact that the formation or disassociation of a deuteron involves the emission or absorption of a gamma ray photon is significant evidence that the states and characteristics of the nucleons in a nucleus are also subject to quantization.

Conclusion

The binding energy of a nuclide is the same as
the potential energy losses due to its formation.