define half-life

Radioactive Decay – Half‑life (Cambridge International AS & A Level Physics 9702)

Learning outcomes

• Explain why radioactive decay is a random (statistical) process.
• Define activity A and decay constant λ, give their units and relationship A = λN.
• Derive the exponential decay law for the number of nuclei N(t) and for activity A(t).
• Define half‑life t_1/2 and show the two equivalent forms

  t_1/2 = \ln 2 / λ and λ = \ln 2 / t_1/2.

• Use the decay law to calculate any of the variables N, A, λ, t_1/2 for a given problem.
• Interpret decay curves, straight‑line (log‑linear) plots and determine t_1/2 experimentally (AO3).
• Design a simple experiment to measure a half‑life, including hypothesis, variables, data‑recording, uncertainty analysis and improvement suggestions.
• State the safety precautions required when handling radionuclides (AO3).
• Connect the energy released in α‑decay (and other modes) to the mass‑defect and nuclear binding‑energy concepts (topic 23.1).
• Recognise related nuclear‑physics ideas required elsewhere in the syllabus: decay series, branching ratios, secular equilibrium, Q‑value calculations, and applications such as medical imaging and radiocarbon dating.

1. Context – where radioactive decay fits in the A‑Level nuclear physics unit

Radioactive decay (topic 23.2) is one part of a larger nuclear‑physics framework. The following checklist shows the neighbouring concepts you will also need to master:

• 23.1 – Mass‑defect, binding energy, Q‑value of nuclear reactions.
• 23.2 – Radioactive decay, decay constant, half‑life, decay series, branching ratios, secular equilibrium.
• 24 – Nuclear reactions, fission, fusion, applications (energy, medicine, dating).
• 22 – Particle physics basics (leptons, quarks, weak interaction) – useful for understanding β‑decay.

2. What is radioactive decay?

Radioactive decay is a spontaneous, random transformation of an unstable nucleus into a more stable configuration. In the process one or more particles (α, β⁻, β⁺) and/or γ‑rays may be emitted. The probability that a particular nucleus will decay in a short time interval dt is constant and independent of the past history of that nucleus.

3. Types of radiation and their characteristics

4. Activity and decay constant

• Decay constant (λ) – probability per unit time that a single nucleus will decay.

  
  Units: s^‑1.

• Activity (A) – number of decays per unit time.

  
  A = λN (where N is the number of undecayed nuclei).

  
  Unit: becquerel (Bq) = 1 decay s^‑1.

Example 1 – Activity from N and λ

Given N = 2.0 × 10¹² and λ = 1.5 × 10^‑4 s^‑1:

A = λN = (1.5 × 10^‑4 s^‑1)(2.0 × 10¹²) = 3.0 × 10⁸ Bq.

5. Exponential decay law

The rate of change of N is proportional to the number present:

\$\frac{dN}{dt} = -\lambda N\$

Integrating with the initial condition N(0)=N₀ gives

\$N(t)=N_{0}\,e^{-\lambda t}\$

Multiplying by λ yields the activity form:

\$A(t)=A{0}\,e^{-\lambda t}\qquad\text{where }A{0}=λN_{0}\$

6. Half‑life

The half‑life t_1/2 is the time required for half of the original nuclei (or half of the activity) to disappear:

\$N(t{1/2})=\frac{1}{2}N{0}\$

Substituting the decay law and solving:

\$\frac{1}{2}N{0}=N{0}e^{-\lambda t{1/2}}\;\Longrightarrow\;e^{-\lambda t{1/2}}=\frac12\$

\$-\lambda t_{1/2}= \ln\!\left(\frac12\right)=-\ln2\$

\$\boxed{t_{1/2}= \frac{\ln2}{\lambda}\approx\frac{0.693}{\lambda}}\$

Re‑arranged:

\$\boxed{\lambda = \frac{\ln2}{t_{1/2}}}\$

Example 2 – Half‑life from decay constant

If λ = 2.5 × 10^‑3 s^‑1:

t_1/2 = 0.693 / (2.5 × 10^‑3) ≈ 2.77 × 10² s ≈ 277 s.

Example 3 – Decay constant from half‑life

Given t_1/2 = 8.0 days:

Convert to seconds: 8.0 days × 86 400 s day^‑1 = 691 200 s.

λ = 0.693 / 691 200 s ≈ 1.0 × 10^‑6 s^‑1.

Example 4 – Remaining nuclei after a given time

A sample contains N₀ = 1.0 × 10¹² nuclei of a radionuclide with t_1/2 = 5 min. How many nuclei remain after 20 min?

First find λ: λ = 0.693 / (5 min × 60 s min^‑1) = 0.00231 s^‑1**.

Then use the decay law:

\$N(20\,\text{min}) = N_{0}e^{-\lambda t}=1.0\times10^{12}\,e^{-0.00231\times1200}\approx1.0\times10^{12}\,e^{-2.77}\approx6.3\times10^{10}\$

7. Decay series, branching ratios and secular equilibrium

• Decay series – many radionuclides decay through a chain of successive α and β decays until a stable nucleus is reached (e.g. the ²³⁸U series). Each member of the series has its own half‑life.
• Branching ratio – some nuclei have two or more possible decay modes. The probability of each mode is expressed as a percentage (e.g. ⁴⁰K decays 89 % by β⁻ and 11 % by electron capture).
• Secular equilibrium – occurs when a long‑lived parent (half‑life ≫) continuously produces a short‑lived daughter. After a few daughter half‑lives the activities become equal: Aparent = Adaughter. This concept is often tested in AO2 questions involving series.

8. Q‑value and mass‑defect

For any nuclear transformation the energy released (Q‑value) is obtained from the mass difference between reactants and products:

\$Q = \left(\sum m{\text{initial}}-\sum m{\text{final}}\right)c^{2}\$

In α‑decay:

\$^{A}{Z}\!X \;\rightarrow\; ^{A-4}{Z-2}\!Y \;+\; ^{4}_{2}\!He \quad\text{(α particle)}\$

The mass of the parent nucleus exceeds the sum of the daughter and α‑particle masses; the “missing” mass is converted into kinetic energy of the emitted particles. A larger Q‑value generally corresponds to a shorter half‑life because the barrier penetration probability is higher.

9. Determining half‑life experimentally (AO3)

Objective: Measure the half‑life of a β‑emitting radionuclide (e.g. ⁶⁰Co) using a Geiger‑Müller (GM) tube.

9.1 Experimental plan

1. Hypothesis: The activity will decrease exponentially with a half‑life equal to the accepted value (5.27 yr for ⁶⁰Co). Over the short laboratory time‑scale the decay will be negligible, but the method can be demonstrated with a short‑lived source such as ²²Na (t_1/2 ≈ 2.6 yr) or a laboratory‑prepared ¹³⁷Cs sample (t_1/2 ≈ 30 yr). For illustration we will use a ⁹⁰Sr source (t_1/2 ≈ 28.8 yr) and record counts for 10 min intervals over several hours.
2. Variables:

   • Independent – elapsed time t (s).
   • Dependent – count rate R (counts s^‑1).
   • Controlled – source‑to‑detector distance, detector voltage, ambient background, temperature.

3. Apparatus:

   • Sealed β source (known activity).
   • GM tube with appropriate β‑shielding.
   • Digital timer / data‑logger.
   • Lead shield to reduce background γ‑rays.
   • Radiation badge for personal monitoring.

4. Method:

   1. Measure background count rate for 5 min; record as B.
   2. Place the source at a fixed distance (e.g. 5 cm) from the GM tube.
   3. Start the timer and record counts for successive 10 min intervals (or any convenient interval). Repeat for at least 8–10 intervals to span > 2 half‑lives of the observed decay (in practice, for a short‑lived laboratory source).
   4. Subtract background from each reading: Ri = (Ci / Δt) – B.

5. Data‑recording table (example):

6. Data analysis:

   • Plot ln R versus time t. The points should lie on a straight line.
   • Determine the slope m by linear regression; m = –λ.
   • Calculate the half‑life: t_1/2 = 0.693 / λ = 0.693 / |m|.
   • Propagate the uncertainty from the slope to t_1/2 using

     \$\frac{σ{t{1/2}}}{t{1/2}} = \frac{σ{λ}}{λ}\$

     where σ_λ is the standard error of the slope.

7. Possible improvements:

   • Increase counting time per interval to reduce statistical (√N) uncertainty.
   • Use a more stable high‑voltage supply for the GM tube.
   • Measure background before and after the experiment to check for drift.
   • Repeat the experiment with a different source to verify the method.

Worked example – extracting t_1/2 from a log‑linear plot

Suppose the linear fit to the data gives a slope m = –2.31 × 10^‑4 s^‑1** with a standard error of 0.05 × 10^‑4 s^‑1. Then:

\$λ = |m| = 2.31\times10^{-4}\,\text{s}^{-1}\$

\$t_{1/2}= \frac{0.693}{2.31\times10^{-4}} = 3.00\times10^{3}\,\text{s} \approx 50\ \text{min}\$

Uncertainty:

\$\frac{σ{t{1/2}}}{t{1/2}} = \frac{σ{λ}}{λ}= \frac{0.05\times10^{-4}}{2.31\times10^{-4}} = 0.022\$

\$σ{t{1/2}} = 0.022 \times 3.00\times10^{3}\,\text{s} \approx 66\ \text{s}\$

Result: t_1/2 = (3.00 ± 0.07) × 10³ s (≈ 50 ± 1 min).

10. Safety and handling of radionuclides (AO3)

• ALARA principle – keep radiation “As Low As Reasonably Achievable”.
• Work with the smallest practical amount of material; maximise distance (inverse‑square law).
• Use appropriate shielding: lead for γ, Plexiglass or thin metal for β, no shielding for α (use a sheet of paper).
• Wear personal protective equipment: lab coat, disposable gloves, safety glasses, and a radiation badge.
• Handle sealed sources with tweezers; never point the source at yourself or others.
• Store sources in labelled, locked containers; keep a written inventory.
• Dispose of radioactive waste according to local regulations; never pour liquids down the sink.
• In case of a spill, evacuate the area, use absorbent material, and follow the institution’s emergency procedure.

11. Applications of half‑life concepts

• Medical imaging and therapy – Technetium‑99m (t_1/2 ≈ 6 h) for diagnostic scans; Iodine‑131 (t_1/2 ≈ 8 days) for thyroid treatment.
• Radiocarbon dating – Carbon‑14 (t_1/2 ≈ 5 730 yr) used to determine the age of archaeological samples.
• Industrial tracers – Use of short‑lived isotopes (e.g. ²⁴Na, t_1/2 ≈ 15 h) to study fluid flow.
• Nuclear power – Knowledge of half‑lives of fission products is essential for waste management.

12. Typical half‑life values

13. Key points to remember

• Activity A = λN; λ is the probability per second that a nucleus decays.
• Exponential decay: N(t)=N₀e^{-λt} and A(t)=A₀e^{-λt}.
• Half‑life t₁/₂ = \ln2 / λ ≈ 0.693/λ (or λ = \ln2 / t₁/₂).
• After each successive half‑life the remaining activity is halved again.
• Decay is random; the statistical uncertainty in a count N is ≈√N (Poisson statistics).
• Decay series, branching ratios and secular equilibrium are essential for multi‑step problems.
• Half‑life can be obtained from a straight‑line plot of ln A versus time, or directly from the time at which the activity falls to half its initial value.
• Safety: minimise exposure, use shielding, wear PPE, monitor with badges, and follow ALARA.
• Energy released (Q‑value) in any decay is linked to the mass‑defect; larger Q‑values generally give shorter half‑lives.
• Applications range from medicine and archaeology to nuclear power and environmental monitoring.