OxfordAQA International AS and A-level Physics: Radioactive decay modes, half-life, radiation properties and the inverse-square law (9630) – Revision Notes

These notes condense section 3.3.3 (Radioactivity) of the OxfordAQA International AS and A-level Physics (9630) specification, version 4.4, for exams from May/June 2020 onwards. The content is shared by International AS and International A-level. Exponential decay equations are International A2 only (section 3.9) and are not covered here. For full explanations use the study guide, then test yourself with the practice questions.

Related pages: the OxfordAQA Physics hub, the printable checklist and the free 10-minute diagnostics.

Decay modes at a glance

Mode	What happens in the nucleus	Change in A	Change in Z	Also emitted
Alpha (α)	Nucleus ejects a helium nucleus	−4	−2	Nothing else
Beta-minus (β⁻)	Neutron becomes proton	0	+1	Electron and antineutrino
Beta-plus (β⁺)	Proton becomes neutron	0	−1	Positron and neutrino
Gamma (γ)	Excited nucleus drops to a lower state	0	0	Photon only

Balance rule: total A and total Z are equal on both sides. Neutrino and antineutrino have A = 0 and Z = 0, so they never change the balance, but they must be included.

Equations to recognise

• Alpha: Ra-226 (Z = 88) → Rn-222 (Z = 86) + α
• Beta-minus: Co-60 (Z = 27) → Ni-60 (Z = 28) + e⁻ + antineutrino
• Beta-plus: C-11 (Z = 6) → B-11 (Z = 5) + e⁺ + neutrino
• Free neutron: n → p + e⁻ + antineutrino
• Parent of technetium-99m: Mo-99 (Z = 42) → Tc-99m (Z = 43) + e⁻ + antineutrino

Method: writing a decay equation

1. Write the parent with A and Z.
2. Choose the particle emitted from the mode named.
3. Subtract A and Z of the emitted particles from the parent to get the daughter.
4. Use the Z to name the element.
5. Check both totals.

The neutrino argument

• Alpha particles from one nuclide have one fixed energy (or a few).
• Beta particles from one nuclide have a continuous range of energies up to a maximum.
• The energy of the beta particle plus the recoil nucleus seemed to be less than the energy released, and momentum seemed not to balance.
• A third, neutral, very weakly interacting particle was hypothesised to carry the missing energy and momentum.

Half-life

Definition: the time taken for half the unstable nuclei in a sample to decay. The activity and corrected count rate halve in the same time.

Half-lives	1	2	3	4	5	6
Fraction left	1/2	1/4	1/8	1/16	1/32	1/64

Rules:

• Decay is random: only the probability for each nucleus is fixed.
• Number of half-lives = elapsed time ÷ half-life.
• Fraction left = (½) to the power of the number of half-lives.
• This spec point uses whole numbers of half-lives.

Method: half-life from a graph

1. Subtract the background rate from every reading.
2. Plot corrected rate against time and draw a smooth curve.
3. Read the time for the rate to halve from the start.
4. Repeat from a second starting value on the curve.
5. Check the values agree, then average.

Small worked reminder. A source has activity 4.8 × 10⁵ Bq and a half-life of 20 min. After 1 h (3 half-lives) the activity is 4.8 × 10⁵ ÷ 8 = 6.0 × 10⁴ Bq.

Gamma and technetium-99m

• After alpha or beta decay the daughter nucleus may be in an excited state, and emits a gamma photon to reach a lower state.
• Gamma emission does not change A or Z: Tc-99m → Tc-99 + γ.
• Technetium-99m is used in medical diagnosis because it emits gamma that leaves the body for detection and does not emit alpha or beta, and its half-life of about 6 hours is long enough for a scan but short enough to limit the dose.

Properties and absorption

	Alpha	Beta	Gamma
Nature	Helium nucleus	Electron or positron	Photon
Ionising	Strongest	Medium	Weakest
Stopped by	Paper, a few cm of air	A few mm of aluminium	Thick lead greatly reduces it

Identification by absorption: measure background first, then corrected count rate with no absorber, paper, aluminium, then lead.

• Drop with paper: alpha present.
• Further drop with aluminium: beta present.
• Only lead reduces what remains: gamma present.

Hazards

• Outside the body: gamma and beta reach tissue, alpha mostly does not penetrate the skin’s outer layer.
• Inside the body: alpha deposits its energy in a tiny volume and is the most hazardous.

Thickness gauging

• Paper, aluminium foil: beta source, a detector opposite, and a count rate that falls as the sheet thickens.
• Steel: gamma source, as beta would be stopped.
• Choose a long half-life so the source does not fade during use.

Inverse-square law (gamma)

I = I₀/r²

• Double the distance: intensity ÷ 4. Triple: ÷ 9.
• Valid for a point source and negligible absorption by air.
• Verify: plot corrected rate against 1/r², expect a straight line through the origin, or check rate × r² is constant.
• Safe handling: tongs, distance, shielding, short exposure, a locked store.

Background radiation

• Origins: radon from rocks, cosmic rays, ground and buildings, food, medical sources, fallout.
• Elimination: count with no source for a long time, take the rate, subtract it from every reading.

Must-know distinctions

• Alpha, beta and gamma: alpha and beta are particles from the nucleus; gamma is a photon. Only gamma leaves A and Z unchanged.
• Neutrino and antineutrino: an electron is emitted with an antineutrino, a positron with a neutrino.
• Activity and count rate: activity is decays per second in the source, in becquerel. A detector records only a fraction of them, plus background.
• Corrected and measured rate: corrected = measured minus background, and only corrected values follow the half-life or inverse-square law.
• Technetium-99m and technetium-99: the m state has extra energy and emits the gamma; the product is the ground state.
• Inside and outside the body: the ranking of hazard reverses.

Worked reminders

Dose activity. A technetium-99m dose is 480 MBq at 06:00. With a half-life of 6.0 h, 18:00 is 12 h later, which is 2 half-lives: 480 ÷ 4 = 120 MBq.

Background. Three five-minute counts with no source: 104, 96 and 112. Mean = 312 ÷ 3 = 104 per five minutes, which is 20.8 per minute. A source reading of 760 per minute becomes 760 − 20.8 = 739 per minute.

Experiment checks. Keep the source-to-detector distance fixed in absorption work, count for the same time each run, and handle sources with tongs.

Quick self-test

1. Write the alpha decay of Ra-224 in terms of A and Z, giving the daughter (Z = 86 is radon).
2. A nucleus emits an antineutrino. Which decay is it, and what changes inside the nucleus?
3. Which emitted particle comes with beta-plus decay besides the positron?
4. Why does a continuous beta energy spectrum suggest another particle is emitted?
5. A sample of sodium-24 has half-life 15 h. What fraction is left after 75 h?
6. A radioisotope’s activity falls from 3200 Bq to 100 Bq in 40 min. What is its half-life?
7. A measured count rate is 130 per minute and the background is 18 per minute. What is the corrected rate?
8. A corrected rate is 150 s⁻¹ at 0.20 m from a gamma source. What is it at 0.50 m?
9. Which of alpha, beta and gamma suits steel thickness gauging, and why?
10. Why is an alpha emitter most dangerous when swallowed?

Answers

1. Ra-224 (A = 224, Z = 88) → Rn-220 (A = 220, Z = 86) + α (A = 4, Z = 2).
2. Beta-minus: a neutron changes into a proton, and an electron is also emitted.
3. A neutrino.
4. Fixed energy release shared by two bodies would give one beta energy. A range means a third particle shares the energy.
5. 75 ÷ 15 = 5 half-lives, so 1/32.
6. 3200 → 100 is a factor of 32, which is 5 half-lives, so 40 ÷ 5 = 8 min.
7. 130 − 18 = 112 per minute.
8. r rises by 2.5, so the rate falls by 6.25: 150 ÷ 6.25 = 24 s⁻¹.
9. Gamma, because it passes through steel and its transmission varies with thickness, while beta is stopped.
10. It is strongly ionising and deposits all its energy in a small volume of living tissue.

Where marks are usually lost

• Leaving the antineutrino or neutrino out of a beta equation, or using the wrong one.
• Writing the electron with a mass number of 1 instead of 0.
• Using the measured rate when the background should be subtracted first.
• Stating half-life as the time for the sample to “half decay”.
• Calling alpha harmless without saying it is outside the body only.
• Using 1/r instead of 1/r² in the inverse-square law.
• Describing an absorption experiment without measuring the background or fixing the geometry.
• Giving a gamma source for foil gauging without justifying the choice.

Official syllabus

OxfordAQA International AS and A-level Physics (9630), specification version 4.4, for exams from May/June 2020 onwards, published by Oxford International AQA Examinations: section 3.3.3 Radioactivity.