Radioactive decay and nuclear instability

Physics 7408 · AQA A Level · 5 min read

Introduction to Radioactive Decay

In this chapter, we explore the "why" and "how" behind radioactive atoms. Some atoms are naturally unstable—you can think of them as being "uncomfortable" because they have too much energy or an awkward balance of particles in their nucleus. To fix this, they spit out radiation to reach a more stable state. This process is called radioactive decay. It is a completely random process, meaning we can't predict exactly when a specific nucleus will decay, but we can use some clever maths to predict what a large group of them will do.

The $N$-$Z$ Graph: The Map of Stability

To understand why an atom is unstable, we look at its $N$-$Z$ graph. This is a plot of the number of neutrons ($N$) against the number of protons ($Z$).

The Stability Belt: Stable nuclei follow a specific pattern. For light elements (up to $Z = 20$), the most stable nuclei have an equal number of protons and neutrons ($N = Z$). As nuclei get heavier, they need more neutrons than protons to stay stable. This is because neutrons act like "nuclear glue" (via the strong nuclear force) to overcome the massive electrostatic repulsion between all those positive protons. The line of stability therefore curves upwards away from the $N = Z$ line.

Decay Modes Based on Position:

Alpha ($\alpha$) decay: Occurs in very heavy nuclei (bottom right of the graph, beyond $Z = 82$). These nuclei are simply too big for the strong nuclear force to hold them together, so they emit an alpha particle (2 protons and 2 neutrons) to slim down.
Beta-minus ($\beta^-$) decay: Occurs in neutron-rich nuclei (above the stability belt). A neutron changes into a proton, moving the nucleus down and to the right toward the stability belt.
Beta-plus ($\beta^+$) decay and Electron Capture: Occurs in proton-rich nuclei (below the stability belt). A proton changes into a neutron, moving the nucleus up and to the left.

Quick Review: If a nucleus is above the line, it has too many neutrons ($\beta^-$). If it is below, it has too many protons ($\beta^+$ or electron capture). If it is way off the top end, it is too heavy ($\alpha$).

Decay Equations and Energy Levels

When a nucleus decays, it often doesn't just change its identity; it also releases energy. We can show these transitions using nuclear energy level diagrams.

Gamma ($\gamma$) Emission and Metastable States

After an alpha or beta decay, the "daughter" nucleus is often left in an excited state—it has excess energy. It releases this energy as a gamma photon to reach its "ground state" (lowest energy). No protons or neutrons are lost during gamma emission, so the element stays the same.

Technetium-99m ($^{99m}\text{Tc}$): The "m" stands for metastable. This is a special version of Technetium that stays in an excited state for a relatively long time (about 6 hours) before emitting a gamma ray. This makes it incredibly useful in medical imaging because it emits only gamma (which passes out of the body easily) and has a short enough half-life to not stay in the patient for too long.

The Mathematics of Decay

Even though individual decays are random, the rate of decay for a large sample is very predictable. This is called exponential decay.

Key Terms:

Activity ($A$): The number of decays per second, measured in Becquerels (Bq). $1 \text{ Bq} = 1 \text{ decay per second}$.
Decay Constant ($\lambda$): The probability of a single nucleus decaying per unit time. It is measured in $\text{s}^{-1}$.
Number of Nuclei ($N$): The total number of undecayed radioactive nuclei remaining in the sample.

The core relationship is: $A = \lambda N$

The Exponential Equation

Because the number of nuclei decreases over time, we use the following equation to find out how many are left after a certain time $t$:

$N = N_0 e^{-\lambda t}$

Since Activity is proportional to $N$, the same pattern applies: $A = A_0 e^{-\lambda t}$

Half-Life ($T_{1/2}$)

The half-life is the average time it takes for the number of undecayed nuclei (or the activity) to halve.

There is a fixed relationship between the half-life and the decay constant:

$T_{1/2} = \frac{\ln 2}{\lambda} \approx \frac{0.693}{\lambda}$

Memory Trick: If a substance has a large decay constant, it is very unstable and decays quickly, meaning it will have a short half-life.

Graphical Analysis

In your exams, you might be asked to find these values from a graph.

Decay Curve ($N$ vs $t$): A curved graph that never quite touches the x-axis. You can find the half-life by picking a starting value, halving it, and seeing how much time has passed on the x-axis.
Logarithmic Graphs: If we take the natural log ($\ln$) of the decay equation, we get a straight line:
$\ln N = -\lambda t + \ln N_0$.
This fits the format $y = mx + c$. If you plot $\ln N$ against $t$, the gradient of the straight line is $-\lambda$.

Applications of Radioactive Decay

Radioactivity isn't just a theoretical concept; it has vital real-world uses.

Radioactive Dating

Carbon-14 dating is used to find the age of once-living materials. While alive, organisms take in Carbon-14. When they die, they stop taking it in, and the Carbon-14 already inside them decays with a known half-life (about 5730 years). By measuring the current activity, we can calculate how long ago the organism died.

Radioactive Waste Storage

Understanding half-life is critical for safety. Some waste products from nuclear reactors have half-lives of thousands of years. These must be stored in stable geological locations (like deep underground in thick concrete bunkers) to ensure they don't leak into the water supply while they are still highly radioactive.

Key Takeaway: Radioactive decay is a random process governed by the decay constant $\lambda$. The $N$-$Z$ graph tells us why a nucleus is unstable, while the exponential decay equations tell us how long it will take to become safe.

(Note: For more details on the properties of Alpha, Beta, and Gamma radiation, or the specifics of Nuclear Fission, see the adjacent chapters in the Nuclear Physics section.)

Quick check

Can you answer these now?

Open each question to check the key ideas from this chapter.

What does it mean for radioactive decay to be a random process?

Radioactive decay is random because it is impossible to predict exactly when a specific nucleus will decay, though the behavior of a large group of nuclei is predictable.

What is the $N$-$Z$ graph and what does it represent?

The $N$-$Z$ graph plots the number of neutrons ($N$) against the number of protons ($Z$) to show the zone of stability for different nuclei.

Which nuclei typically undergo alpha ($\alpha$) decay and why?

Alpha decay occurs in very heavy nuclei (usually $Z > 82$) because they are too large for the strong nuclear force to remain effective against electrostatic repulsion.

Under what conditions do Beta-minus ($\beta^-$) and Beta-plus ($\beta^+$) decay occur?

$\beta^-$-decay occurs in neutron-rich nuclei (above the stability belt), while $\beta^+$-decay occurs in proton-rich nuclei (below the stability belt).

Why is Gamma ($\gamma$) radiation often emitted following alpha or beta decay?

Gamma emission occurs when a "daughter" nucleus is left in an excited state and releases excess energy to reach its ground state.

What is a metastable state, and why is Technetium-99m useful in medicine?

A metastable state is an excited nuclear state that lasts for a relatively long time; $^{99m}\text{Tc}$ is useful because it emits only gamma rays and has a short half-life.

Define Activity ($A$) and the Decay Constant ($\lambda$).

Activity is the number of decays per second (measured in Becquerels, Bq); the decay constant is the probability of a single nucleus decaying per unit time.

What is the equation for the exponential decay of a radioactive sample?

$N = N_0 e^{-\lambda t}$ or $A = A_0 e^{-\lambda t}$.

What is the relationship between half-life ($T_{1/2}$) and the decay constant ($\lambda$)?

$T_{1/2} = \frac{\ln 2}{\lambda} \approx \frac{0.693}{\lambda}$.

How can the decay constant ($\lambda$) be determined from a logarithmic graph?

By plotting $\ln N$ against time ($t$), the gradient of the resulting straight line is equal to $-\lambda$.

How does Carbon-14 dating determine the age of organic materials?

It measures the remaining activity of Carbon-14 in a dead organism and compares it to the activity while alive to calculate the time elapsed since death.

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