Nuclear Forces, Defect of Mass and Binding Energy Revision Notes

In these revision notes for Nuclear Forces, Defect of Mass and Binding Energy, we cover the following key points:

• What is nuclear force? Where does it act? What kind of force is it (attractive or repulsive)?
• What force does balance the nuclear force?
• What is mass-energy equivalence? When does it occur?
• What is defect in mass? How to calculate it?
• What is binding energy? How to calculate it?
• How does the number of neutrons N vary with the atomic number Z in various elements?
• What is the excess of neutrons? When does it occur?

Nuclear Forces, Defect of Mass and Binding Energy Revision Notes

Since in any atomic nucleus there are Z positively charged protons, there is an attracting force that keeps the nucleons together, as opposed to repelling electric force caused by like charged protons. This force known as nuclear force, must balance the electric force acting between protons. In order to perform its function (i.e. to keep nucleons inside the nuclei), nuclear force must act in very short distances, not more than 10^-15 m, as it must not exceed the dimensions of atomic nuclei.

The two forces acting between nucleons: electric (repulsive) and nuclear (attractive) vary with the distance of nucleons from each other. This distance determines which force overcomes the other. Logically, there must be an equilibrium position in which the nucleus acquires stability and where the magnitudes of the above two forces are equal (the net force is zero).

The effect of (repelling) electric force increases with the increase in the atomic number Z. For this reason, elements heavier than Uranium (Z > 92) manifest lack of stability in their nuclei and tend to break down in various ways. Hence, they are known as unstable chemical elements. The stability of nuclei depends on many other factors besides these ones however.

Although the total mass of a system may change, the total energy and momentum remain constant. This statement is known as the mass-energy equivalence law and it was discovered by Einstein. The corresponding equation is

In previous articles, we have called this energy "rest energy". In suitable conditions, it can be entirely converted into mass (matter) and vice-versa. However, this is a very complicated process.

Nucleons cannot be considered as free particles as they are interacting inside the nucleus by means of nuclear forces caused by the nuclear field, which makes them possess nuclear energy. Therefore, a nucleus with A nucleons possesses energy in two forms: in the mass of nucleons and energy of nuclear field.

From the law of energy conservation and the mass-energy equivalence, we expect a difference in mass between the sum of free individual nucleons and the mass of nucleus made by the same nucleons. Such a difference in mass is also confirmed experimentally. In scientific terms, it is known as "mass defect" or "binding energy", depending on the approach. From the law of energy conservation, it is clear that binding energy, E_b is calculated through the equation:

where m_p is the mass of proton, m_n the mass of neutron and M is the mass of the whole nucleus. Simplifying both sides by c², we obtain the formula for the mass defect:

The system is bound when the mass defect is positive (Δm > 0), i.e. when the mass of nucleus is smaller than the sum of masses of the individual nucleons.

The average binding energy per nucleon Eb(A) is a function of atomic mass A. Binding energy shows how stable the nucleus is. The higher the binding energy the more stable a nucleus is. For two isobars (Z is different and A is equal), the nucleus with the lowest atomic number Z (the one with the highest N therefore) is more stable. This is because such nuclei contain less protons and as a result, the electric force between them is smaller. Hence, the nuclear force needed to balance the system is smaller as well.

In general, N > Z. This fact is known as excess of neutrons.