E.3.1 Stability and binding energy Notes

Understanding Isotopes, Nuclear Stability, and the Strong Nuclear Force

1. Every atom is defined by its number of protons, denoted as $Z$, which determines the element.
2. However, not all carbon atoms are identical, they can have different numbers of neutrons, $N$.
3. The total number of protons and neutrons, called the nucleon number, is denoted as $A$, where: $$
   A = Z + N
   $$
4. Atoms of the same element with different numbers of neutrons are called isotopes.

Key Properties of Isotopes

1. Same chemical properties: Since chemical behavior depends on the number of electrons, and isotopes have the same $Z$, their chemistry is identical.
2. Different physical properties: The difference in neutron number affects properties like mass and nuclear stability.

Binding Energy Per Nucleon: A Measure of Stability

When you assemble protons and neutrons to form a nucleus, the total mass of the nucleus is slightly less than the sum of the individual masses of its protons and neutrons.

This "missing mass" is called the mass defect, and it is converted into energy according to Einstein’s famous equation: $$
E = mc^2
$$

This energy, known as the binding energy, is what holds the nucleus together. The binding energy per nucleon is a key indicator of nuclear stability.

$$
\text{Binding Energy per Nucleon} = \frac{\text{Total Binding Energy}}{\text{Number of Nucleons}}
$$

The Binding Energy Curve

If we plot the binding energy per nucleon against the nucleon number $A$, we observe:

1. Rapid increase for small nuclei: Light nuclei like helium ($A = 4$) have relatively low binding energy per nucleon.
2. Peak stability near iron ($A \approx 60$): Iron-56 has one of the highest binding energies per nucleon, making it extremely stable.
3. Gradual decrease for heavier nuclei: As $A$ increases beyond 60, the binding energy per nucleon decreases, making larger nuclei like uranium less stable.