AQA A-level Physics · 7408 · Nuclear physics

Radioactivity: decay, activity and half-life

Turn random nuclear events into predictable decay equations, then analyse detector counts, logarithmic graphs and gamma inverse-square data.

8 worked examplesRandom-decay explorerRequired Practical 1242 marks of practice

The same fraction remains after equal time intervals. This smooth curve describes the expectation for a population.

Start with a two-minute check

A detector records 60 counts in 20 s with a source present. Background alone gives 24 counts in 40 s. What is the source-only count rate?

Your prediction

Convert each count to a rate before subtracting background.

Three passes through the topic

Understand: read radiation properties, random decay and the difference between activity and count rate.
Calculate: work through the examples, then use the explorer and practical data to test the equations.
Retrieve: answer the short checks and 42-mark question set before opening the answers.

By the end, you should be able to

Compare alpha, beta and gamma radiation and explain how absorption identifies them.
Explain spontaneous, random decay with constant probability per nucleus.
Correct detector data for background and distinguish count rate from activity.
Calculate activity from nuclei or mass, half-life and decay constant.
Use exponential equations, half-life steps and natural-log graphs.
Test the gamma inverse-square law and evaluate Required Practical 12.
Explain thickness monitoring, dating, waste storage and radiation risk/benefit.
Explain why small random samples fluctuate around a smooth decay curve.

Alpha, beta and gamma radiation: properties

Ionisation is the removal of electrons from atoms. Ionising radiation can damage cells by ionising atoms and molecules. The following comparisons describe typical school-level sources; penetration depends on radiation energy and the absorbing material.

Three common types of nuclear radiation
Radiation	Nature and charge	Relative ionisation	Penetration and absorption
Alpha, α	Helium nucleus: 2 protons + 2 neutrons; charge +2e; mass about 4 u	Strong over a short path	Short range in air; stopped by paper or the outer skin
Beta, β⁻	Fast electron emitted from the nucleus; charge −e; very small mass	Intermediate	Passes through paper; typically stopped by a few mm of aluminium
Gamma, γ	High-energy electromagnetic photon; no charge or rest mass	Less ionising per unit path than typical alpha/beta emissions	Very penetrating; intensity is reduced by thick lead or concrete

Charged alpha and beta particles can be deflected by electric and magnetic fields; gamma is undeflected. A β⁻ electron is created in a nuclear transformation, rather than being an orbital electron knocked out of the atom. β⁺ radiation consists of positrons, with charge +e and the same mass as an electron.

Nuclear changes: alpha emission reduces mass number by 4 and proton number by 2. Beta-minus emission keeps mass number constant and raises proton number by 1. Gamma emission reduces nuclear energy without changing either number. Revisit particles and radiation for balanced nuclear equations and conservation laws.

Why is radioactive decay random?

Decay is spontaneous: an unstable nucleus decays without an external trigger. It is random: you cannot predict which nucleus will decay next or the exact time for a particular nucleus. For a given isotope, each undecayed nucleus has the same constant probability of decaying in a specified short time interval, regardless of its age.

This does not mean that a source loses the same number of nuclei each second. As the number remaining decreases, fewer nuclei are available to decay. A large population gives an approximately smooth exponential curve, while a small population shows obvious statistical fluctuations.

Probability in an interval Δt = 1 − e^−λΔt ≈ λΔt when λΔt is small

For a dice model with removal probability p per throw, the expected survivors after k throws are N₀(1 − p)^k. The number removed varies randomly. Throws are a discrete model; do not treat a large per-throw probability as an exact continuous decay constant.

Ordinary changes in temperature, pressure or chemical state do not appreciably change the decay constant of the isotopes in these A-level models.

Background radiation and corrected count rate

Background comes from cosmic radiation, radioactive rocks and soil, radon gas, building materials and naturally occurring isotopes in food and the body. Its detector reading fluctuates because decay and detection events are random.

r_gross = C_{source present}/t_{source present}
r_background = C_background/t_background
r_net = r_gross − r_background

C is a recorded count, t is the counting time and r is count rate in s⁻¹ (counts per second). Subtract raw counts directly only if their counting times match. Measure background over a long interval with the source stored well away, using the same detector settings.

Worked example 1 · unequal counting times

Background gives 720 counts in 300 s. With the source present the detector records 1860 counts in 60 s.

Background rate = 720/300 = 2.40 s⁻¹.
Gross rate = 1860/60 = 31.0 s⁻¹.
Corrected rate = 31.0 − 2.40 = 28.6 s⁻¹.

The result is a detector rate. It is not automatically an activity of 28.6 Bq: the detector usually registers only a fraction of the source's decays.

Why count longer, and what does a negative corrected reading mean?

For independent random counts, the approximate standard deviation of a large count C is √C. Its fractional uncertainty is about 1/√C, so longer counting generally reduces relative uncertainty. Uncertainties in the gross and background rates both contribute to the corrected rate. A small negative result after subtraction can occur through statistical fluctuations; it does not mean the source has negative activity. It indicates a signal too small to distinguish reliably with those measurements.

Activity, decay constant and number of nuclei

Activity A is the rate of nuclear decay in a source. Its unit is the becquerel: 1 Bq = one decay per second. The decay constant λ is the probability per unit time of decay for an undecayed nucleus in the short-interval limit; its SI unit is s⁻¹.

A = λN and N = (m/M)N_A

N is the number of undecayed radioactive nuclei, not the nucleon number of one nucleus. For a pure sample of that isotope, m/M is the amount in moles and N_A = 6.022 × 10²³ mol⁻¹. Mass m and molar mass M must use matching units. If only a fraction of a sample is the radioactive isotope, use that isotope's mass.

With fixed geometry, absorption, detector efficiency and emission branching, the background-corrected count rate is proportional to activity: r_net = εA, where ε is the mean number of registered events per decay in that setup. It is generally less than one for a detector viewing a small solid angle. Moving the detector changes ε and hence the measured count rate even if the source's activity hardly changes.

Worked example 2 · activity from a tiny mass

A pure isotope sample has mass 1.8 pg, molar mass 90 g mol⁻¹ and half-life 6.0 h. Find its initial activity.

Convert mass: m = 1.8 × 10⁻¹² g. Then N = (1.8 × 10⁻¹²/90) × 6.022 × 10²³ = 1.2044 × 10¹⁰ nuclei.
T_½ = 6.0 × 3600 = 21600 s; λ = ln 2/21600 = 3.209 × 10⁻⁵ s⁻¹.
A = λN = 3.865 × 10⁵ Bq ≈ 0.39 MBq.

A small mass can contain a very large number of nuclei. A and the nucleon number use the same letter in different contexts; read the units and definition.

Radioactive decay equations and half-life

For an isotope with constant λ, the expected rate of change of its undecayed population is proportional to the number still present:

dN/dt = −λN (ΔN/Δt ≈ −λN for a short interval)
N = N₀e^−λt and A = A₀e^−λt

The minus sign describes the decrease in N. Activity is a positive rate: A = −dN/dt = λN. The same exponential applies to background-corrected count rate when the detection arrangement stays fixed. If a daughter isotope also contributes to the measured radiation, a single exponential may not describe the total detected rate.

Half-life T_½ is the time for the expected undecayed population, or its activity, to halve. Substitute N/N₀ = 1/2 into the exponential:

T_½ = ln 2/λ = 0.693/λ
A/A₀ = N/N₀ = (½)^t/T½
t = ln(A₀/A)/λ

Fractions remaining after complete half-lives
Half-lives elapsed	Fraction remaining	Percentage remaining
0	1	100%
1	1/2	50%
2	1/4	25%
3	1/8	12.5%
4	1/16	6.25%

Worked example 3 · three half-lives

A source starts at 2.4 MBq and has half-life 3.0 h. What is its activity after 9.0 h?

λ = ln 2/(3.0 × 3600) = 6.418 × 10⁻⁵ s⁻¹. Three half-lives have elapsed, so A = 2.4/2³ = 0.300 MBq.

Alternatively, use A = 2.4e^{−(ln 2/3.0)×9.0} MBq with λ in h⁻¹. Both calculations use consistent time units.

Worked example 4 · a non-integer number of half-lives

Activity falls from 80 kBq to 6.0 kBq. The half-life is 4.0 h. Find the time taken.

t = ln(80/6.0)/(ln 2/4.0) = 14.948 h ≈ 15 h.

Check: four half-lives would take 16 h and leave 5.0 kBq. Reaching 6.0 kBq must take slightly less time.

Unit check: λt must be dimensionless. Convert days or hours to seconds when λ is in s⁻¹. Activity ratios can use kBq/kBq; do not mix kBq with Bq in the same ratio.

Decay curves and logarithmic graphs

On an ordinary N–t graph the curve becomes less steep. Its tangent gradient is dN/dt, so the magnitude of that gradient is activity. Measure the time to halve from several different starting levels; an exponential has the same half-life throughout.

Taking natural logarithms gives a straight-line relationship. Divide the count rate by a reference rate so the logarithm has a dimensionless argument:

ln[r_net/(1 s⁻¹)] = ln[r₀/(1 s⁻¹)] − λt
Natural-log gradient = −λ

Fit a straight line through the trend and use a large triangle on that line. If time is in minutes, the gradient gives λ in min⁻¹. If the vertical axis uses log₁₀ instead, its gradient is −λ/ln 10; it is not −λ.

Natural logarithm of corrected count rate against time in minutes, falling in a straight line with gradient minus 0.040 per minute. — Natural-log ordinates are about 5.298 at 0 min, 4.978 at 8 min and 4.658 at 16 min. The corresponding corrected rates are 200, 145.23 and 105.46 s⁻¹.

Worked example 5 · half-life from a log graph

The best-fit natural-log line falls by 0.640 over 16.0 min.

Gradient = −0.640/16.0 = −0.0400 min⁻¹.
λ = 0.0400 min⁻¹ = 6.67 × 10⁻⁴ s⁻¹.
T_½ = ln 2/0.0400 = 17.3 min.

Logarithmically spaced graph paper: if the labels say N = 10, 100, 1000, they are values of N, even though equal spacing represents multiplication by ten. For a natural-log gradient use ln N at each point, not the raw difference in N. Correct background before taking logs; zero or negative corrected readings cannot be logged.

The area under an activity–time graph is the expected number of decays during that interval: ∫A dt = N_start − N_end. An area under a detector-rate graph gives expected detected counts, which usually represent only a fraction of those decays.

Random-decay explorer: one run versus the expectation

This model samples an independent, exponentially distributed lifetime for every nucleus. Each decays once into a stable daughter. It represents all decays in the sample, with no detector losses or background. The half-life is an illustrative model value, rather than a named isotope's half-life.

Initial undecayed nucleiModel half-life

Explore the sampled run: 0.0 s

Use the slider or arrow keys to inspect the same run from 0 to 5 half-lives. Changing the population or half-life generates a new run.

For 200 nuclei with half-life 4 s: λ = 0.1733 s⁻¹ and initial expected activity = 34.66 Bq. Use the static expected-value table below if the controls are unavailable.

One sampled runSmooth expectation

The solid stepped curve is one population; the dashed curve is the expectation. The vertical marker follows the time slider.

Bars show observed decays divided by the bin duration (half a half-life). Dashed marks show the expected mean rate in each bin. A bin average is different from instantaneous activity λN.

Accessible data for this random run

Observed and expected mean decay rates by time interval
Interval / s	Decays	Observed / s⁻¹	Expected / s⁻¹
The random-run table appears when JavaScript runs. The expected-value table below is always available.

Static expectation: N₀ = 200, T½ = 4 s
Time / s	Expected nuclei	Expected activity / Bq
0	200	34.66
4	100	17.33
8	50	8.66
12	25	4.33
16	12.5	2.17
20	6.25	1.08

Predict → observe → explain

Choose 20 nuclei and inspect one half-life. Is exactly 10 always left? Generate several new runs and explain the differences.
Choose 2000 nuclei. Compare fractional fluctuations with those for 20. The expectation is unchanged as a fraction of the initial population.
Double the half-life at the same initial N. Predict how λ and initial activity change, then check the readout.
Find two adjacent bars where the observed mean rate rises. Explain how that can coexist with a falling smooth expectation.

Answers to the explorer tasks

No: 10 is the expected number. Each nucleus decays at a random time, so one run may differ.
The larger population normally has smaller relative fluctuations; both have the same expected fraction remaining at a given number of half-lives.
λ = ln 2/T½ halves, so λN₀ and the initial expected activity halve.
Counts fluctuate randomly from bin to bin. The population N cannot rise, but a later bin can contain more observed events than an earlier one.

For a single nucleus, open the Particle & Atom Lab, choose Atom Builder, build or choose an unstable isotope, predict the daughter and use Run clock. Its screen clock is accelerated. One nucleus's sampled lifetime is not the same as the half-life of a population.

Absorption experiments and thickness monitoring

Measure the corrected rate with no absorber, then insert paper, thin aluminium and thick lead while keeping source–detector geometry and counting time fixed. A large fall with paper suggests alpha; further loss with aluminium suggests beta; radiation transmitted through both but attenuated by lead suggests gamma. Real sources can emit a mixture, and material thickness and radiation energy matter.

Compare differences against random counting uncertainty. If the source has appreciable decay over the investigation, account for the time trend rather than attributing every fall to the absorber.

Paper or thin aluminium: beta gauge

Place the sheet between a fixed beta source and detector. Select energy and geometry so some beta passes through. A thicker sheet absorbs more radiation and reduces the corrected detector rate; a thinner sheet increases it. A feedback system can adjust the rollers to maintain a target rate.

Alpha is usually stopped by paper and would give little useful transmitted signal. Gamma may pass too readily through a thin sheet to provide sufficient sensitivity.

Thicker steel: gamma gauge

Gamma is more penetrating and can be partly transmitted through thicker metal. With suitable energy and detector arrangement, increasing thickness reduces the measured rate.

The useful operating range is partial transmission: a detector reading already close to zero cannot sensitively distinguish a small additional thickness.

Attenuated is not completely stopped. Gamma intensity falls as absorber thickness increases; thick shielding does not imply that every photon is absorbed. Distinguish absorption by matter from geometric spreading with distance.

The gamma inverse-square law

For an approximately point-like source emitting in all directions, the same power spreads over spheres of area 4πx². With negligible absorption between source and detector:

I = k/x² and r_net ∝ 1/x²
r_net,2/r_net,1 = (x₁/x₂)²

I is intensity (power per unit area), whereas r is detector count rate. At fixed detector response the corrected count rate follows the same distance dependence. Use the distance from the active source to the detector window, not a gap measured from the outside of its holder.

Worked example 6 · doubling distance

At 0.15 m, a gamma source gives net rate 120 s⁻¹. Background is 4.0 s⁻¹. What gross rate is expected at 0.30 m?

Net rate at 0.30 m = 120(0.15/0.30)² = 30.0 s⁻¹.
Gross rate = 30.0 + 4.0 = 34.0 s⁻¹.
The original gross rate was 124 s⁻¹; dividing that by four gives 31, which incorrectly quarters the background too.

An ideal corrected-rate check
x / m	rnet / s⁻¹	rnet x² / s⁻¹ m²
0.15	120	2.70
0.30	30.0	2.70
0.45	13.33	2.70 (rounded)

The constant product supports inverse-square dependence. Use every supplied row and allow for rounding and uncertainty. This model is unsuitable when the source or detector dimensions are comparable with x, or when significant attenuation occurs. Do not apply it blindly to alpha over distances comparable with its range.

AQA Required Practical 12: gamma inverse-square investigation

A sealed gamma source in a holder faces an aligned detector connected to a counter. A ruler measures distance x from the active source to the detector window. — Schematic, not to scale. Keep source, window and ruler aligned. Identify the reference points used to measure x.

Background: with the source in its appropriate shielded store, count background over a long measured time; calculate its rate.
Geometry: mount the sealed gamma source and detector securely, with the detector facing the source. Measure x from active source to detector window. Keep detector settings and alignment fixed.
Counts: choose several distances, sufficiently larger than source/detector dimensions. Record counts for a measured interval at each distance, repeat and average. Longer counting is especially useful at larger distances.
Correction: convert each count to a rate, then subtract background. Record x, time, gross rate, background rate, corrected rate and uncertainties.
Analysis: plot r_net vertically against 1/x² horizontally. An approximately straight line through the origin supports the law. Alternatively compare r_netx² across the full data set, or use a log–log graph with expected gradient −2.
Evaluation: compare deviations with uncertainties and identify a physical cause with its likely effect.

Safe handling: follow the teacher's authorised procedure for the sealed source. Use long handling tongs, keep the exposure brief and maximise distance; return the source to its appropriate shielded store when it is not needed. Keep hands away from the source: moving the detector to vary x can reduce unnecessary handling. Time, distance and shielding each reduce exposure.

Explain specific limitations

Background: at large x, the source signal may be small relative to background; subtraction then leaves a large relative uncertainty.
Distance offset: measuring from a holder edge gives the wrong x. This matters particularly at small distances because of the square dependence.
Finite source or detector: at short x the point-source/small-detector approximation is less accurate.
Dead time: a detector may fail to register closely spaced arrivals after an event. High rates can be underestimated, making the close-distance points lie below the expected trend. Random events can arrive close together even when their average spacing is longer.
Counting fluctuations: repeated readings differ through statistical randomness; longer counts and repeats improve precision.

A non-zero intercept needs interpretation: residual background or another systematic effect may remain. “Human error” alone is not an explanation. If background was already corrected, do not subtract it again or claim an uncorrected background explains the supplied corrected values.

Radioactive dating and waste-storage calculations

Dating compares the current parent population or activity with its initial value. That initial value needs evidence or an assumption. In a closed parent–daughter system with one stable daughter per decay and no daughter initially, N₀ = N_parent + N_{radiogenic daughter}.

Worked example 7 · reconstruct the original population

A mineral has 4.0 × 10⁸ parent nuclei and 1.2 × 10⁹ radiogenic daughter nuclei. The parent's half-life is 6.0 × 10⁸ years.

N₀ = 4.0 × 10⁸ + 1.2 × 10⁹ = 1.6 × 10⁹.
N/N₀ = 1/4, so two half-lives have passed.
Age = 2 × 6.0 × 10⁸ = 1.2 × 10⁹ years.

This assumes a closed system, no initial daughter and one stable daughter per parent decay. If those conditions fail, this simple estimate is not valid.

For carbon dating, living material exchanges carbon with its environment. After death that exchange stops; the carbon-14 population decreases. Age estimates compare its activity with an appropriate initial reference, allowing for assumptions about the original carbon-14 abundance.

Waste storage often involves estimating how long activity takes to fall below a stated threshold. A long half-life means the isotope persists; a short half-life can mean high initial activity for the same N. Neither half-life alone nor a small sample mass establishes safety.

Worked example 8 · time to a stated threshold

An isolated parent isotope has half-life 18 days. Activity starts at 4.8 MBq; a calculation asks when it falls below 30 kBq.

Use the same activity units: A₀/A = 4800/30 = 160. Then t = 18 ln(160)/ln 2 = 131.8 days at the threshold, so it is below it for times greater than this.

If checking only at complete half-lives, seven leave 37.5 kBq and eight leave 18.75 kBq. The first complete-half-life check below 30 kBq is 144 days.

The ideal exponential never reaches exactly zero. Real storage decisions also account for daughter isotopes, containment and other hazards; a question's threshold is a stated modelling condition.

Radiation hazards and medical risk–benefit

The hazard depends on radiation type, energy, activity, exposure time, distance and whether the source is outside or inside the body. Irradiation means exposure to radiation; contamination means radioactive material is deposited on or inside something.

External and internal exposure

Alpha is usually stopped by the outer skin, so an external alpha source is less hazardous than the same material inhaled or ingested. Inside the body, its strong ionisation can damage nearby living tissue. Beta can affect skin and internal tissue. Gamma is penetrating and can irradiate organs from outside the body.

There is no single radiation type that is “always the most dangerous”. State the location and exposure conditions.

Useful applications have risks

Medical tracers can reveal organ function; imaging can identify conditions needed for treatment; radiotherapy can damage cancer cells. Ionising radiation can also damage healthy tissue. A justified procedure weighs the diagnostic or treatment benefit against that risk and limits unnecessary exposure.

Gamma can sterilise packaged equipment because it penetrates the packaging and damages microorganisms. In ordinary gamma sterilisation, ionisation does not make the equipment's nuclei unstable. Irradiation and radioactive contamination are different processes.

Eight quick retrieval checks

What do “random” and “spontaneous” each mean?
Define activity and state the meaning of one becquerel.
Why does activity decrease if each remaining nucleus keeps the same decay probability?
How do you correct counts measured over unequal times?
State the half-life equation and the gradient of a natural-log decay plot.
Which absorber helps distinguish alpha from beta, and which material strongly attenuates gamma?
Why can an alpha source be more hazardous inside the body?
State a graph and a measurement precaution for Required Practical 12.

Show retrieval answers

Random: a particular decay time cannot be predicted. Spontaneous: no external trigger is required.
Activity is the rate of nuclear decay in a source; 1 Bq is one decay per second.
There are fewer undecayed nuclei remaining; A = λN falls while λ stays constant.
Divide each count by its own time, then subtract background rate from gross rate.
T½ = ln 2/λ; a natural-log corrected-rate versus time graph has gradient −λ.
Paper stops typical alpha while allowing beta through. Thick lead or concrete strongly attenuates gamma.
Inside, it can reach living tissue and ionise strongly; externally the outer skin usually stops it.
Plot corrected rate against 1/x². Measure x from active source to window, keep alignment fixed and correct background; a relevant stated precaution is accepted.

Common radioactivity mistakes and exam checks

Subtracting counts from different intervals: calculate rates first.
Calling detector rate the source activity: consider efficiency, geometry and background.
Using nucleon number as N: N counts radioactive nuclei in the sample.
Expecting exactly half of a small sample to remain: half-life is a statistical expectation.
Subtracting a fixed number each half-life: halve the current number each time.
Using −λ for a base-ten logarithm: check which logarithm the axis uses.
Mixing time units: make λt dimensionless before using exp.
Quartering the gross rate at double distance: apply inverse square to the source-only part, then add background.
Using current parent + initial parent as the original total: in the stated dating model add current parent + radiogenic daughter.
Claiming radiation always makes an object radioactive: distinguish ionisation from nuclear change and contamination.
Explaining disagreement vaguely: calculate the difference and connect a specific limitation to its effect.

Before finishing a calculation: write the equation, show conversions and substitution, retain intermediate digits, give units and check whether the answer should rise or fall. For explanations, connect the physics to the actual material, detector or exposure in the question.

Original AQA radioactivity exam practice · 42 marks

These are original practice questions covering calculations, graphs, practical reasoning, explanation and a mixed decay–power problem. Try them before opening the answers. Extra digits in the worked mark points help you check intermediate steps; round final answers sensibly.

1. Background correction [3 marks]

A detector records 900 background counts in 600 s. With a source present it records 2400 counts in 120 s. Calculate the background rate, gross rate and source-only count rate.