IB DP Physics: Activity, half-life and the radioactive decay law (DP Physics) – Revision Notes

These notes condense the second part of sub-topic E.3, Radioactive decay, in the IB Physics guide (first assessment 2025): activity, count rate, half-life, background radiation, finding a half-life and real-life uses. For full explanations and longer worked examples, read the study guide, then test yourself with the practice questions. Everything applies to SL and HL, except items marked “HL only” (the decay constant, the exponential law and arbitrary time intervals). Types of decay and binding energy are in the separate unit. Links: course hub, printable checklist, free 10-minute diagnostics.

Definitions

Term	Meaning
Random	Which nucleus decays next, and when, cannot be predicted
Spontaneous	Decay is not triggered by outside conditions
Activity, A	Number of nuclei decaying per unit time; unit becquerel, Bq = one decay per second
Count rate	Counts registered by the detector per unit time; less than the activity
Background radiation	Counts recorded with no source present (radon, rocks, cosmic rays, buildings, food, medical sources)
Corrected count rate	Measured count rate minus background count rate
Half-life, T½	Time for the number of undecayed nuclei (and the activity) to fall to half
Decay constant, λ (HL only)	Constant in A = λN; unit s⁻¹ or per hour, day, year

Method: whole numbers of half-lives (SL and HL)

1. Find n = t / T½. Check that it is a whole number.
2. Divide the starting value by 2ⁿ.
3. If background is involved, subtract it first, divide, then add it back to get a reading.

n	1	2	3	4	5	6
Fraction left	1/2	1/4	1/8	1/16	1/32	1/64
Percentage left	50%	25%	12.5%	6.25%	3.125%	1.56%

Worked reminder. A source has A₀ = 1280 Bq and T½ = 20 min. After 1.0 h, n = 60/20 = 3, so A = 1280/8 = 160 Bq.

The same factor applies to N, A and corrected count rate. Half-life does not depend on the amount of sample, so it takes just as long to halve from 100 nuclei as from 10⁶ nuclei (on average).

Background radiation

• Measure it with the source removed, over a long time, and calculate a rate.
• Subtract it from every reading before you compare, plot or halve.
• Do not halve the background. Only the source’s contribution decays.
• Counts are random, so a longer count time gives a smaller fractional uncertainty.

Method: determining a half-life

Step	What to do
1	Measure background rate with the source far away, over a long time
2	Fix source and detector so the geometry stays the same
3	Count in equal intervals at regular times
4	Subtract background from each rate
5	Plot corrected rate against time and draw a smooth curve
6	Find the time to halve from several starting values, then average
HL extra	Plot ln(corrected rate) against time; gradient = −λ; T½ = ln2/λ

Safety: handle sources with tongs, keep them in a shielded store when not in use, keep your distance, and limit exposure time.

Real-life contexts

Context	Radiation	Half-life	Reason
Medical imaging	Gamma	Short (hours)	Gamma leaves the body to the camera; the dose fades quickly
Medical treatment	Usually beta, taken up by the target tissue	Hours to days	Energy is delivered over a short range, inside the target
Pipe leak tracer	Gamma	Short	Gets through soil and pipe wall; does not linger in the ground
Thickness gauging	Beta for thin sheet, gamma for thick	Long	Radiation must be partly absorbed; the signal must stay steady
Dating	Decay of a long-lived isotope	Comparable to the age	Enough, but not too few, nuclei remain to measure

Dating method. Divide the activity per gram of living material by the activity per gram of the sample. Write the ratio as 2ⁿ. Age = n × T½. Assume the starting activity per gram was the same as in living material today.

Alpha radiation is stopped by paper or a few centimetres of air, so it is no use for gauging or tracing through material, and it is hazardous if the source is inside the body.

HL only: the decay law

Quantity	Equation
Number of nuclei	N = N₀ e^(−λt)
Activity	A = λN = λN₀ e^(−λt) = A₀ e^(−λt)
Half-life link	T½ = ln2 / λ
Time to reach a fraction f	t = ln(1/f) / λ
Two readings	λ = ln(A₁/A₂) / (t₂ − t₁)

Method for any time interval.

1. Convert to one time unit. If A is wanted in Bq, put λ in s⁻¹.
2. Find λ from λ = ln2 / T½ (or from two readings).
3. Put λt in the exponent. Do not round λ too early.
4. Check against whole half-lives: t/T½ close to a whole number should give about (½)ⁿ.

Probability meaning. The probability that a given nucleus decays in a short interval Δt is approximately λΔt, but only when λΔt is much less than 1. The exact value is 1 − e^(−λΔt).

Worked reminder. Carbon-14 has T½ = 5730 years, so λ = ln2/5730 = 1.21 × 10⁻⁴ year⁻¹. A sample with a quarter of the original activity is at n = 2, so its age is 11 460 years. For a sample at 35% of the original, t = ln(1/0.35)/λ = 8.7 × 10³ years.

Must-know distinctions

• Activity versus count rate. Activity is the decays in the source; count rate is what the detector registers.
• Measured versus corrected rate. Halve only the corrected rate.
• Half-life versus decay constant. A large λ means a short T½.
• Half-life versus lifetime. After two half-lives a quarter of the nuclei remain, not none.
• Random versus unpredictable in bulk. One nucleus is unpredictable; a large sample follows a smooth curve.
• Approximation versus exact. λΔt is a probability only when it is small.

Quick self-test

1. Define the activity of a source and state its unit.
2. A source has activity 7.2 × 10⁶ Bq and half-life 12 days. Find its activity after 36 days.
3. How long does it take for the activity to fall from 6400 Bq to 400 Bq if T½ = 5.0 min?
4. A detector reads 75 counts per minute with a source and 15 counts per minute without. T½ = 4.0 h. What does it read 8.0 h later?
5. What percentage of a radioactive sample remains after four half-lives?
6. The activity of a sample falls from 800 Bq to 100 Bq in 9.0 days. Find the half-life.
7. Explain why an alpha emitter is a poor choice of tracer for a leak in an underground pipe.
8. State two reasons why the count rate is smaller than the activity.
9. HL only: a nuclide has T½ = 2.0 h. Calculate λ in h⁻¹.
10. HL only: λ = 0.025 s⁻¹. Calculate the half-life.
11. HL only: a sample starts with 5.0 × 10⁶ undecayed nuclei and λ = 0.020 min⁻¹. How many remain after 30 min?
12. HL only: a sample holds 2.0 × 10¹² nuclei and λ = 4.0 × 10⁻⁶ s⁻¹. Find its activity.

Answers

1. The number of nuclei that decay per unit time. Unit: becquerel (Bq), one decay per second.
2. n = 36/12 = 3; A = 7.2 × 10⁶ / 8 = 9.0 × 10⁵ Bq.
3. 6400 → 3200 → 1600 → 800 → 400 is four halvings. Time = 4 × 5.0 = 20 min.
4. Corrected rate = 75 − 15 = 60. n = 2, so 60/4 = 15. Reading = 15 + 15 = 30 counts per minute.
5. (½)⁴ = 1/16 = 6.25%.
6. 800/100 = 8 = 2³, so three half-lives in 9.0 days, so T½ = 3.0 days.
7. Alpha radiation is absorbed by soil and the pipe wall, so it never reaches a surface detector.
8. Any two of: the source emits in all directions, so the detector covers only part of them; some radiation is absorbed on the way (air, source or window); the detector does not register every particle.
9. λ = ln2 / 2.0 = 0.35 h⁻¹ (0.347).
10. T½ = ln2 / 0.025 = 28 s (27.7 s).
11. N = 5.0 × 10⁶ × e^(−0.020 × 30) = 5.0 × 10⁶ × e^(−0.6) = 2.7 × 10⁶.
12. A = λN = 4.0 × 10⁻⁶ × 2.0 × 10¹² = 8.0 × 10⁶ Bq.

Where marks are usually lost

• Halving the measured count rate, not the corrected count rate.
• Forgetting to add the background back when the question asks for the reading.
• Writing n = t × T½ instead of t / T½.
• Leaving out units, or giving the becquerel as “counts per second” for a detector reading.
• Justifying the isotope with only the type of radiation, and not its half-life.
• Describing a half-life experiment with no background measurement or with no repeated halving times.
• Saying a nucleus “knows” when to decay, or that decay depends on temperature.
• HL: using minutes for t with λ in s⁻¹.
• HL: rounding λ to one figure before it enters the exponent.
• HL: using λΔt as a probability when it is not small.

Official syllabus

IB Diploma Programme Physics guide, first assessment 2025, published by the International Baccalaureate Organization. Sub-topic E.3, Radioactive decay.