IB Physics E.3.3 Quantitative Analysis of Decay Notes

Radioactive Decay Law and Related Concepts

1. While holding a sample of radioactive material in your hand, you know its nuclei are unstable. At some point, they will decay, releasing particles or energy.
2. But how can you predict how much of it will remain after a certain time? How can the rate of decay be quantified, and how does this relate to the material's half-life?

These questions are at the heart of the radioactive decay law, a cornerstone of nuclear physics.

Evidence for the Strong Nuclear Force

1. While holding a sample of radioactive material, you might wonder what holds its nucleus together despite the repulsive forces between protons.
2. The answer lies in the strong nuclear force, which is responsible for binding protons and neutrons (nucleons) within the nucleus.
3. Unlike the electrostatic force, which decreases with distance according to an inverse-square law, the strong force is short-ranged and attractive, acting only at distances of about $10^{-15}$ m (1 femtometer).

The Role of the Neutron-to-Proton Ratio in Nuclear Stability

1. Not all combinations of protons and neutrons form stable nuclei.
2. The stability of a nuclide depends on the neutron-to-proton (N/Z) ratio.
3. Light elements (such as carbon and oxygen) tend to have nearly equal numbers of protons and neutrons ($N \approx Z$), while heavier elements require more neutrons than protons to remain stable.
4. This increasing neutron excess is necessary to counteract the growing electrostatic repulsion between protons.

Approximate Constancy of the Binding Energy Curve for $A > 60$

1. As discussed in the earlier sections, the binding energy per nucleon determines the stability of a nucleus.
2. For small nuclei, adding more nucleons significantly increases the binding energy per nucleon, making them more stable.
3. However, beyond a nucleon number of around 60 (near iron-56) the binding energy per nucleon reaches its maximum and then decreases only gradually, so it stays nearly constant over a wide range of heavier nuclei.

This behavior is explained by the balance between the strong nuclear force (which is short-ranged) and the electrostatic repulsion (which acts over longer distances).

As the nucleus gets larger, nucleons interact mostly with their immediate neighbors rather than with all nucleons in the nucleus, leading to the plateau in the binding energy curve.

The Radioactive Decay Law

Radioactive decay is a random and spontaneous process.