Master exponential decay, activity calculations, decay constants, and half-lives for nuclear physics.
AQA A-Level Physics · Section 8 · Nuclear Physics
⚛️
Define activity A and state its units (Bq)
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Use A = λN and λ = ln2 / t½
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Apply N = N₀e^(−λt) and A = A₀e^(−λt)
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Analyse decay graphs (linear and logarithmic)
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Use Geiger-Muller tube measurements correctly
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Verify the inverse-square law for gamma radiation
Activity and the Decay Constant
Radioactive decay is a random and spontaneous process. The number of decays per second is called the activity A, measured in becquerels (Bq) where 1 Bq = 1 decay per second.
A = λN
Where:
A = activity (Bq)
λ = decay constant (s⁻¹) — probability of decay per unit time
N = number of undecayed nuclei
The decay constant λ is the probability that an individual nucleus will decay per unit time. It is a fixed property of each isotope and is unaffected by physical or chemical conditions.
Half-life and Decay Constant
The half-life t½ is the time taken for the number of undecayed nuclei (or the activity) to halve. The relationship between λ and t½ is:
λ = ln 2 / t½ t½ = ln 2 / λ = 0.693 / λ
Half-life and decay constant are inversely proportional: a short half-life means a large decay constant and vice versa.
Isotope
Half-life
Use
Carbon-14
5730 years
Archaeological dating
Iodine-131
8 days
Medical thyroid treatment
Technetium-99m
6 hours
Medical imaging (gamma)
Uranium-238
4.5 × 10⁹ years
Geological dating
Exponential Decay Equations
The number of undecayed nuclei and the activity both decrease exponentially:
N = N₀ e^(−λt) A = A₀ e^(−λt)
Taking the natural log of the activity equation:
ln A = ln A₀ − λt
This is the equation of a straight line: a plot of ln A vs t gives:
Gradient = −λ
y-intercept = ln A₀
Plotting ln(count rate) vs time allows the decay constant (and hence half-life) to be found experimentally from the gradient.
Geiger-Muller Tube Measurements
A Geiger-Muller (GM) tube detects ionising radiation. Key considerations:
Background radiation must be measured and subtracted from all readings
Dead time: the GM tube cannot detect a second pulse immediately after one — it recovers in ~200 μs
Count rate (counts per second or per minute) is proportional to activity
GM tubes have different efficiencies for α, β, and γ radiation
Always subtract background count rate before plotting decay graphs or applying the inverse-square law. Failing to do so is a common source of experimental error.
Inverse-Square Law for Gamma Radiation
Gamma radiation spreads out in all directions from a point source. The intensity (and count rate) follows an inverse-square law:
I ∝ 1 / d² or I₁d₁² = I₂d₂²
To verify experimentally:
Measure corrected count rate C at several distances d from the source
Plot C vs 1/d² — should give a straight line through the origin
Or plot ln C vs ln d — gradient should equal −2
The inverse-square law applies to gamma radiation (not alpha or beta, which are absorbed by air at short distances). It arises because the same energy spreads over a sphere of area 4πd² as distance increases.
Example 1: Finding activity from number of nuclei
A sample contains 3.0 × 10¹⁵ nuclei of iodine-131 (t½ = 8.04 days). Calculate the decay constant and the initial activity.
1 Convert half-life to seconds: t½ = 8.04 × 24 × 3600 = 6.95 × 10⁵ s
A GM tube records a corrected count rate of 400 s⁻¹ at 0.20 m from a gamma source. What count rate is expected at 0.50 m?
1 Inverse-square law: C₁d₁² = C₂d₂²
2 C₂ = C₁ × (d₁/d₂)² = 400 × (0.20/0.50)²
3 C₂ = 400 × (0.40)² = 400 × 0.16
C₂ = 64 s⁻¹
Q1. Which of the following correctly defines the decay constant λ?
Q2. A source has a half-life of 20 s and an initial activity of 1600 Bq. What is the activity after 1 minute?
Q3. Why must background radiation always be subtracted in GM tube experiments?
Q4. If the decay constant of a sample is λ = 0.050 s⁻¹, what is the half-life?
Q5. A student plots ln(count rate) against time and obtains a straight line with gradient −0.023 min⁻¹. What is the half-life?
Challenge 1. A freshly prepared sample of phosphorus-32 (t½ = 14.3 days) has an activity of 5.0 × 10⁶ Bq. Calculate (a) λ in s⁻¹, (b) the number of P-32 nuclei in the sample, and (c) the mass of P-32 (molar mass 32 g/mol, Nₐ = 6.02 × 10²³ mol⁻¹).
Challenge 2. A GM tube records a count rate of 350 s⁻¹ at 0.10 m from a gamma source and 46 s⁻¹ at 0.40 m. The background count rate is 30 s⁻¹. Verify whether the inverse-square law holds.
Corrected count rates: C₁ = 350 − 30 = 320 s⁻¹; C₂ = 46 − 30 = 16 s⁻¹
Check C₁d₁² = C₂d₂²:
320 × (0.10)² = 320 × 0.010 = 3.20
16 × (0.40)² = 16 × 0.16 = 2.56
These are not exactly equal (3.20 vs 2.56), giving about 20% discrepancy.
This suggests some absorption by air or departure from a true point source, or experimental uncertainty in distance measurement. The trend is broadly consistent with an inverse-square law but discrepancies highlight the importance of background subtraction and precise distance measurement.
Challenge 3. Carbon dating: A wooden artefact contains 12.5% of the C-14 of living wood. The half-life of C-14 is 5730 years. Calculate the age of the artefact and explain why C-14 dating is limited to objects younger than about 50,000 years.
12.5% = (1/2)³, so 3 half-lives have elapsed.
Age = 3 × 5730 = 17 190 years
For objects older than ~50,000 years (≈ 8-9 half-lives), the remaining C-14 drops to less than 0.2% of the original.
Count rates become indistinguishable from background radiation, making accurate measurement impossible with a GM tube.
Modern accelerator mass spectrometry can extend this to ~60,000 years by counting individual C-14 atoms rather than waiting for decays.
Required Practical
Geiger-Muller Tube & Inverse-Square Law for Gamma Radiation
Aim
To measure the count rate from a gamma source at various distances and verify that count rate C ∝ 1/d².
Apparatus
Geiger-Muller (GM) tube connected to a counter/ratemeter
Gamma source (e.g. Cs-137 or Co-60) in a sealed holder
Metre rule or optical bench
Lead shielding (for handling source safely)
Stopwatch
Gamma sources must be handled with tongs. Keep sources at arm's length and minimise exposure time. Never point towards people. Follow local radiation safety rules.
Method
Measure and record the background count rate with no source present (count for at least 2 minutes to reduce statistical uncertainty).
Place the GM tube and source on the optical bench. Measure distance d from source to GM tube window.
Record count for a fixed time (e.g. 60 s) at each distance: 0.05, 0.10, 0.15, 0.20, 0.25, 0.30 m.
Repeat each measurement at least twice to reduce random error and average.
Subtract background count rate from each reading to obtain the corrected count rate C.
Analysis
Plot corrected count rate C (y-axis) against 1/d² (x-axis).
If the inverse-square law holds, the graph should be a straight line through the origin.
Alternatively, plot ln C vs ln d — the gradient should be −2.
C = k / d² → ln C = ln k − 2 ln d
Sources of Error & Improvements
Error
Improvement
Random fluctuations in count (statistical)
Count for longer time; repeat measurements
Background radiation varies
Measure background multiple times and average
Distance measured to wrong part of GM tube
Always measure to the GM tube window, not the body
Source not a true point source
Use small sources; work at distances >> source diameter
Absorption of gamma by air
Work in open lab; note that gamma has low absorption anyway