Radioactive Decay & Half-Life
No one can say when a particular unstable nucleus will decay — but a large enough sample lets you predict, with real precision, what fraction remains after an hour, a year, or a millennium. This lesson turns that statistical regularity into the exponential decay law, and shows how it's measured using a Geiger–Müller tube.
Radioactive decay is random and spontaneous
Radioactive decay is random: it's impossible to predict which individual nucleus in a sample will decay next, or exactly when. It's also spontaneous: decay is completely unaffected by external conditions — temperature, pressure, chemical bonding, or nearby nuclei all leave the decay rate unchanged, unlike a chemical reaction, whose rate depends strongly on these factors.
What can be predicted, with high precision, is the behaviour of a large sample. Each nucleus has the same fixed probability of decaying in any given interval, and with enough nuclei, this translates into a smooth, statistical decline in the number remaining over time.
Activity and the decay constant
The activity, $A$, of a sample is the number of nuclear decays occurring per unit time, measured in becquerels (Bq), where $1\,\text{Bq} = 1$ decay per second. Activity is directly proportional to the number of undecayed nuclei present, $N$:
where $\lambda$ is the decay constant, a fixed probability per unit time that any one nucleus of that isotope will decay, with units of $\text{s}^{-1}$. A larger $\lambda$ means a higher probability of decay per second, and so — for the same $N$ — a higher activity: $\lambda$ is a fixed property of the isotope, unlike $N$, which continuously falls as the sample decays.
The exponential decay law
Because activity ($A=\lambda N$, the rate at which nuclei are lost) is itself proportional to the number of nuclei remaining, the rate of decrease of $N$ is proportional to $N$ itself — precisely the condition that produces exponential decay.
The same exponential form applies to activity and to any measured count rate, since both are proportional to $N$: $A = A_0 e^{-\lambda t}$, where $A_0$ is the initial activity. This is convenient, since activity (or count rate, which is proportional to it) can be measured directly, whereas the number of undecayed nuclei usually can't.
Half-life
The half-life, $T_{1/2}$, is the time for the number of undecayed nuclei (or the activity) of a sample to fall to half its initial value — a fixed property of a given isotope, related directly to the decay constant.
Worked example: using the decay law
Choosing a half-life for the job
Half-life isn't just an abstract number to calculate — in practice, an application is only workable if the isotope used has a half-life suited to it. A source decaying too quickly is gone before it's useful; one decaying too slowly stays hazardous for far longer than necessary. The two cases below show how deliberately short and deliberately long half-lives are each put to use, and why a very long half-life brings its own distinct danger.
Short half-lives: medical diagnosis
A radioactive tracer used to image an organ needs to be active for long enough to complete the scan, but should then decay away quickly, so the patient isn't left carrying a source of radiation dose for any longer than necessary. This means diagnostic tracers are deliberately chosen to have short half-lives — long enough (typically several hours) to inject the tracer, let it accumulate in the target organ, and take the images, but short enough that the patient's activity has fallen to a small fraction of its starting value within a day or two.
Long half-lives: dating and long-term standards
The opposite choice is needed when an isotope's decay is used to measure how much time has passed since some fixed starting point — this requires a half-life comparable to, or longer than, the age being measured, or the sample would have decayed away completely long ago. Radiocarbon dating of organic archaeological material relies on carbon-14, with a half-life of about 5,730 years, well matched to timescales of thousands of years; dating much older rock relies on isotopes with half-lives of billions of years, such as uranium-238.
The danger of very long half-lives
Because $A=\lambda N$ and $\lambda = \dfrac{\ln 2}{T_{1/2}}$, a longer half-life means a smaller decay constant, and so — for the same number of nuclei — a lower activity. This can seem reassuring, but it's exactly what makes very long half-lives hazardous in a different way: the low activity doesn't fall away on any useful timescale. A highly active, short-half-life source is dangerous now but safe soon; a long-half-life source stays dangerous for a very long time, since its activity barely decreases from one human lifetime to the next.
This is precisely the problem posed by high-level radioactive waste from nuclear reactors, some of which contains isotopes with half-lives of thousands to millions of years. Such waste can't simply be left to decay away in storage on any practical timescale, and if it escapes containment, it can enter the environment and the food chain — where, once absorbed into a living organism, it continues irradiating that organism from within for as long as it remains present. This is why long-half-life waste requires deep geological storage, engineered to remain sealed for timescales far beyond any human institution's normal lifespan.
It's tempting to think a lower activity always means "safer", but activity only tells you the rate of decay right now — it says nothing about how long that rate will persist. A short half-life gives high activity that disappears quickly; a long half-life gives low activity that persists for a very long time. Judging real hazard means considering both the activity and the half-life together, not the activity alone.
Measuring count rate with a Geiger–Müller tube
A Geiger–Müller (GM) tube detects individual ionising particles: each one entering the tube triggers a brief pulse of current, registered as one "count" by a connected counter. Because only a fraction of the radiation emitted by a source enters the tube (radiation is emitted in all directions, and the tube covers only a small solid angle), the tube's count rate is proportional to, but numerically smaller than, the source's true activity.
A GM tube also detects background radiation — the small, steady rate of ionising events from cosmic rays, naturally occurring isotopes in rocks and building materials, and trace radioactivity in the body, present everywhere regardless of the source studied. Before measuring with a source present, background count rate must be measured separately (with no source nearby) and subtracted from every reading, to obtain the corrected count rate due to the source alone:
Every count-rate measurement in this topic — decay-constant experiments, half-life determinations, the inverse-square-law practical below — depends on the corrected count rate, not the raw reading. Skipping the subtraction systematically overestimates the true source count rate, and the error grows the weaker or more distant the source is, since background then makes up a larger fraction of the raw reading.
Required Practical: Inverse Square Law for Gamma Radiation
This is AQA's Required Practical 12: investigating how radiation intensity varies with distance.
A gamma source radiates energy uniformly in all directions from a point, so the same total energy spreads over the surface of an ever-larger sphere as distance from the source increases. Since a sphere's surface area is proportional to $x^2$ ($x$ = distance from the source), gamma intensity follows an inverse square law:
and since a GM tube's corrected count rate is proportional to the intensity reaching it, count rate should also obey $\text{count rate} \propto \dfrac{1}{x^2}$.
Equipment
| Equipment | Purpose |
|---|---|
| Gamma radioactive source (sealed, low-activity) | Provides a known point source of gamma radiation |
| Tongs | Handles the source without direct contact |
| Geiger-Müller (GM) tube and counter | Detects and counts incident gamma radiation |
| Metre ruler | Measures the distance $x$ between source and tube |
| Source-holder stand | Holds the source securely in line with the ruler and tube |
| Lead-lined storage container | Safely stores the source when not in use |
Method
Sources of error and how to minimise them
Background must be measured and subtracted at every distance, not just once — a poorly corrected background disproportionately distorts readings far from the source, where the true count rate is smallest and background makes up the largest fraction of the reading.
Radioactive decay is random, so a short counting interval gives a count rate with large statistical scatter. Count for a sufficiently long time at each distance (and/or repeat and average) to reduce the percentage uncertainty.
A brief interval after each detected particle, during which the tube can't register a further count, causes under-recording — significant close to the source, where count rates are highest. Avoid working too close to a strong source, or apply a dead-time correction if known.
Distance should be measured from the actual point of emission inside the source to the active detecting volume inside the tube, not from either outer casing — both typically add a few millimetres to a centimetre, introducing a systematic error that matters most at the smallest distances.
Handle sources with tongs at all times, minimise time near the source, maximise distance when not taking a reading, store in a lead-lined container between uses, and never point it toward another person.
Key takeaways
- Radioactive decay is random (unpredictable for any one nucleus) and spontaneous (unaffected by external physical or chemical conditions).
- Activity $A=\lambda N$, where $\lambda$ is the decay constant — a fixed probability of decay per unit time for a given isotope.
- The decay law, from $\dfrac{dN}{dt}=-\lambda N$: $N=N_0e^{-\lambda t}$, and equivalently $A=A_0e^{-\lambda t}$ for activity or count rate.
- Half-life $T_{1/2} = \dfrac{\ln 2}{\lambda}$: the time for $N$ or $A$ to halve, fixed for a given isotope.
- Applications need a half-life matched to their timescale: short half-lives (e.g. Tc-99m, ≈6 hours) suit medical diagnosis, since activity clears quickly after use; long half-lives (e.g. carbon-14, uranium-238) suit dating, since the isotope must survive over the age being measured.
- The molybdenum–technetium generator solves Tc-99m's supply problem: longer-lived Mo-99 (≈66 hours) is transported to the hospital, then "milked" to extract fresh, short-lived Tc-99m on-site as it's needed.
- A very long half-life means low activity now, but that activity barely decreases over practical timescales — the danger of long-lived radioactive waste, requiring long-term (often deep geological) storage.
- A GM tube's raw count rate must be corrected for background radiation (measured separately, with no source present) before it can be used as a proxy for source activity.
- RP12: corrected GM-tube count rate from a gamma source plotted against $1/x^2$ gives a straight line through the origin, confirming $I \propto 1/x^2$; sources of error include background correction, GM tube dead time at high count rates, and measuring distance from the true point of emission, not the outer casings.