Radioactive Decay – Cambridge IGCSE/A‑Level (9702)
Learning Objective
Students will be able to explain why the inevitable fluctuations in measured count rates are quantitative evidence of the random (stochastic) nature of radioactive decay, and to use this understanding in practical investigations.
1. Fundamental Definitions
- Decay constant, λ – probability per unit time that a single nucleus will decay (units s⁻¹).
- Half‑life, t½ – time required for half of the nuclei in a sample to decay.
- Activity, A – number of decays per unit time (Bq). Two equivalent forms are required by the syllabus:
- \(A = \lambda N\)
- \(A = \frac{0.693\,N}{t{½}}\) (because \(\lambda = 0.693/t{½}\))
- Number of undecayed nuclei, N(t) – follows the exponential law
\[
N(t)=N{0}\,e^{-\lambda t}=N{0}\,2^{-t/t_{½}} .
\]
The second form follows directly from the definition of half‑life (see derivation below).
2. Derivation of the Decay Law (Syllabus Requirement)
For a small time interval dt, the probability that a particular nucleus decays is λ dt. If at time t the sample contains N(t) nuclei, the expected number that decay in dt is λ N(t) dt. Hence
\[
-\frac{dN}{dt}= \lambda N .
\]
Integrating with the condition N(0)=N_{0} gives
\[
N(t)=N_{0}e^{-\lambda t}.
\]
Setting t = t{½} gives \(N(t{½}) = N_{0}/2\). Substituting into the exponential expression yields
\[
\frac{1}{2}=e^{-\lambda t{½}}\;\Longrightarrow\;\lambda t{½}= \ln 2\approx0.693,
\]
so that \(\lambda =0.693/t_{½}\). Re‑substituting this value of λ into the exponential gives the alternative half‑life form
\[
N(t)=N{0}\,2^{-t/t{½}} .
\]
3. Random (Stochastic) Nature of Decay
- Each nucleus decays independently; the chance of decay does not depend on what other nuclei are doing.
- The decay of a large number of nuclei is a Poisson process. For a fixed counting time Δt the expected (mean) number of counts is
\[
\langle n\rangle =A(t)\,\Delta t =\lambda N(t)\,\Delta t .
\]
- For a Poisson distribution the variance equals the mean:
\[
\sigma^{2}= \langle n\rangle ,\qquad
\sigma =\sqrt{\langle n\rangle },\qquad
\frac{\sigma}{\langle n\rangle }=\frac{1}{\sqrt{\langle n\rangle }} .
\]
- Consequences for the laboratory:
- Low count rates → large fractional uncertainty.
- Doubling the counting time (or using a stronger source) roughly halves the relative uncertainty.
- Observed fluctuations that obey the Poisson relation are direct evidence of the random nature of decay.
4. Practical Quantities and Units
| Quantity | Symbol | Unit | Typical school‑lab range |
|---|
| Decay constant | λ | s⁻¹ | 10⁻⁴ – 10⁻¹ s⁻¹ |
| Half‑life | t½ | s, min, h, yr | seconds to millions of years |
| Activity | A | Bq (kBq, MBq) | 10 – 10⁶ Bq for typical school sources |
| Count rate | R | counts s⁻¹ | 5 – 500 cps with a GM tube |
5. Accounting for Background and Detector Efficiency
6. Experimental Demonstration of Random Decay
- Set up a Geiger–Müller (GM) tube connected to a digital counter.
- Place a weak radioactive source at a fixed, reproducible distance from the detector.
- Measure the background count rate B for at least 5 min and record the average.
- Choose a counting interval Δt (e.g., 10 s). Record the number of counts \(n_i\) in successive equal intervals for a total of N intervals (N ≥ 30 is recommended).
- Calculate:
- Mean raw count: \(\displaystyle \langle n{\text{raw}}\rangle =\frac{1}{N}\sum{i=1}^{N} n_i\)
- Mean background in the same interval: \(\displaystyle \langle n_{B}\rangle = B\,\Delta t\)
- Net mean count: \(\displaystyle \langle n\rangle =\langle n{\text{raw}}\rangle -\langle n{B}\rangle\)
- Standard deviation of the raw data: \(\displaystyle \sigma{\text{raw}} =\sqrt{\frac{1}{N-1}\sum{i=1}^{N}(ni-\langle n{\text{raw}}\rangle )^{2}}\)
- Poisson prediction: \(\displaystyle \sigma_{\text{Poisson}} =\sqrt{\langle n\rangle}\)
- Compare σraw with σPoisson. Agreement (within experimental error) confirms the stochastic nature of decay.
7. Sample Data Set (Δt = 10 s)
| Interval | Raw counts \(n_i\) |
|---|
| 1 | 12 |
| 2 | 9 |
| 3 | 15 |
| 4 | 11 |
| 5 | 13 |
| 6 | 10 |
| 7 | 14 |
| 8 | 12 |
| 9 | 8 |
| 10 | 13 |
Calculations (background B ≈ 0.2 cps):
- Mean raw count: \(\langle n_{\text{raw}}\rangle = 11.7\) counts per 10 s.
- Background contribution in 10 s: \(\langle n_{B}\rangle = 0.2\times10 = 2\) counts.
- Net mean count: \(\langle n\rangle = 11.7 - 2 = 9.7\) counts.
- Measured standard deviation: \(\sigma_{\text{raw}} \approx 2.1\) counts.
- Poisson prediction: \(\sigma_{\text{Poisson}} = \sqrt{9.7} \approx 3.1\) counts.
With a larger data set (e.g., N = 100) the measured σ approaches the Poisson value, reinforcing the random‑decay model.
8. Using Decay Equations for Predictions
Example – remaining activity after a given time
- Initial activity: \(A_0 = 200\;\text{kBq}\).
- Half‑life: \(t_{½}= 2.5\;\text{h}\).
- After \(t = 7.5\;\text{h}\) (three half‑lives):
\[
A(t)=A0\,2^{-t/t{½}} = 200\;\text{kBq}\times2^{-3}=25\;\text{kBq}.
\]
This shows how a fundamentally random microscopic process yields a deterministic macroscopic exponential decrease.
9. Safety Guidelines (Essential for All Practical Work)
- Handle sources with tweezers or forceps; never touch them with bare hands.
- Store sources in labelled, sealed containers when not in use.
- Maintain a minimum distance of 30 cm between source and detector unless the experiment specifically requires a closer geometry.
- Wear lab coat and safety glasses at all times.
- Record the activity, isotope, and date for every source; follow your institution’s radiation‑safety policy for storage and disposal.
- In the event of a spill or breakage, evacuate the area, inform the supervisor, and follow the approved contamination‑control procedure.
10. Classroom Activities
- Computer simulation: Write a short Python (or Excel) program that generates random numbers to mimic the decay of \(N_0\) nuclei. Plot the simulated \(N(t)\) together with the theoretical exponential curve.
- Monte‑Carlo counting: Students draw random numbers to simulate a 10‑second counting interval, repeat 50 times, and construct a histogram. The histogram should approximate a Poisson distribution.
- Data‑analysis challenge: Provide a mixed data set (different sources, varying background). Students must subtract background, correct for detector efficiency, and determine the half‑life from a series of measurements using the equation \(A = A0 2^{-t/t{½}}\).
11. Key Take‑aways
- Radioactive decay is fundamentally random; each nucleus has a fixed probability λ to decay per unit time.
- The number of decays recorded in a fixed interval follows a Poisson distribution, with variance equal to the mean.
- Statistical fluctuations \(\sigma \approx \sqrt{\langle n\rangle}\) provide quantitative evidence of this randomness.
- Accurate experimental work requires subtraction of background, correction for detector efficiency, and adherence to strict safety procedures.
- Understanding count‑rate fluctuations underpins many applications, from radiometric dating to medical imaging and radiation protection.