Radioactive Decay Law and Related Concepts
- While holding a sample of radioactive material in your hand, you know it’s unstable at some point, its nuclei will decay, releasing particles or energy.
- 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
- While holding a sample of radioactive material, you might wonder what holds its nucleus together despite the repulsive forces between protons.
- The answer lies in the strong nuclear force, which is responsible for binding protons and neutrons (nucleons) within the nucleus.
- 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).
- Experimental evidence for this force comes from nucleon scattering experiments, where high-energy protons are fired at atomic nuclei.
- These experiments reveal a strong, attractive force at very short distances that overcomes the proton-proton electrostatic repulsion.
- Without the strong nuclear force, atomic nuclei would be unable to exist due to the overwhelming repulsive force between protons.
The strong nuclear force is the fundamental force that holds the nucleus together, preventing protons from repelling each other.
The Role of the Neutron-to-Proton Ratio in Nuclear Stability
- Not all combinations of protons and neutrons form stable nuclei.
- The stability of a nuclide depends on the neutron-to-proton (N/Z) ratio.
- 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.
- This increasing neutron excess is necessary to counteract the growing electrostatic repulsion between protons.
- If a nucleus has too few neutrons, it undergoes beta-plus ($\beta^+$) decay to convert a proton into a neutron.
- If it has too many neutrons, it undergoes beta-minus ($\beta^-$) decay to convert a neutron into a proton.
- Extremely unstable nuclei may undergo alpha decay or fission to regain stability.
Nuclei with an unstable neutron-to-proton ratio decay via beta decay to move toward a more stable configuration.
Approximate Constancy of the Binding Energy Curve for $A > 60$
- As discussed in the earlier sections, the binding energy per nucleon determines the stability of a nucleus.
- For small nuclei, adding more nucleons significantly increases the binding energy per nucleon, making them more stable.
- However, beyond a nucleon number of around 60 (approximately iron-56), the binding energy per nucleon remains nearly constant, meaning additional nucleons do not significantly increase nuclear stability.
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.
Common Mistake- Many students assume adding nucleons always increases binding energy per nucleon.
- In reality, the binding energy curve plateaus above $A = 60$, making elements like iron the most stable.
The Radioactive Decay Law
Radioactive decay
Radioactive decay is the process by which an unstable atomic nucleus loses energy by radiation.
Radioactive decay is a random and spontaneous process.Tip
- Random means you cannot predict when a specific nucleus will decay.
- Spontaneous means the decay occurs independently of external factors like temperature or pressure.
- However, for a large number of nuclei, the decay behavior can be described statistically.
- The radioactive decay law is expressed mathematically as: $$\frac{\mathrm{d}N}{\mathrm{d}t} = -\lambda N$$ where:
- $N$ is the number of undecayed nuclei at time $t$,
- $\lambda$ is the decay constant, representing the probability of decay per unit time,
- $\frac{\mathrm{d}N}{\mathrm{d}t}$ is the rate of decay (negative because $N$ decreases over time).
- Solving this differential equation yields the exponential decay formula: $$N = N_0 e^{-\lambda t}$$ where:
- $N_0$: The initial number of nuclei at $t = 0$,
- $t$: The elapsed time.
This equation shows that the number of undecayed nuclei decreases exponentially over time, a hallmark of radioactive decay.
You start with $N_0 = 1000$ radioactive nuclei, and the decay constant is $\lambda = 0.1 \, \mathrm{min}^{-1}$.
How many nuclei remain after 30 minutes?
Solution
- Using the decay law:
$$N = N_0 e^{-\lambda t} = 1000 e^{-0.1 \times 30}$$
$$N = 1000 e^{-3} \approx 1000 \times 0.0498 \approx 49.8$$
- After 30 minutes, approximately 50 nuclei remain.
Decay Constant $\lambda$
Decay constant
The decay constant, $\lambda$, measures how quickly a radioactive substance decays. It is defined as the probability of decay per unit time for a single nucleus.
