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# Nuclear binding energy

## Nuclear binding energy

The nuclear binding energy of a nucleus is the minimum energy required to break the nucleus down into its individual nucleons (i.e. protons and neutrons).

Binding energy in a nucleus is the result of the interplay between two forces: the attractive strong nuclear force (one of the fundamental forces) that binds the nucleons and the repulsive electromagnetic force between the positively charged protons.

The binding energy per nucleon is the binding energy of the nucleus divided by the number of nucleons in the nucleus. It indicates how strongly each nucleon is bound to the rest of the nucleus. Nuclei with higher binding energies per nucleon are more stable.

The changes in the binding energy per nucleon explain why energy can be gained from splitting very large atoms (nuclear fission) and by merging very small atoms (nuclear fusion).

Intuitively, the binding energy per nucleon is equal to the average energy required for a nucleon to break away from a nucleus. This is similar to the gravitational potential energy corresponding to the energy needed for one star to break away from a star cluster.

## Mass defect

The mass defect is the difference between the total mass of the individual nucleons (when they are NOT bound together in a nucleus) and the mass of the nucleus as a whole:$$\delta=\text{ total mass of nucleons} - \text{mass of nucleus}$$

When nucleons are not bound into an atom, they are in a higher energy state than when they are part of an atom. Part of the mass-energy possessed by the nucleons when they are isolated is used to bind them together in a nucleus.

The relationship between binding energy and mass defect builds on Einstein's mass-energy equivalence:$$E_{\text{binding}}=\delta\times c^{2}$$

The mass defect has been empirically measured. Physicists calculated the sum of the mass of protons and neutrons in an atom and measured the mass of the atom itself which was found to be different from the sum.

## Nuclear stability

For stable atoms lighter than iron (i.e. with atomic mass $\ce{A}\lt 56$), the binding energy per nucleon generally increases with increasing nucleon number.

For stable atoms heavier than iron (i.e. $\ce{A}\gt 56$) the binding energy per nucleon decreases with increasing nucleon number of the nucleus.

## Factors affecting nuclear stability

The pattern of nuclear stability arises from the interaction of the strong nuclear force between all nucleons and the electrostatic force between the protons.

The strong force is attractive but only effective over short distances while the electrostatic force is repulsive, weaker and acts over both short and long distances.

The repulsion between protons can be kept in check by adding more neutrons to bind with the protons. Protons are unable to bind with each other as their mutual repulsion is greater than the strong attractive force.

This is why stable atoms with more than 12 protons tend to have more neutrons than protons.

Neutrons, however, also need to be close to protons to be stable (otherwise they will decay into protons). As a result, adding too many neutrons also destabilises the atom.