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Required Practical 7 (RP7) on the AQA A-Level Physics specification is the experimental investigation of the inverse-square law for gamma radiation. Students measure the count rate of gamma radiation as a function of the source-to-detector distance, subtract background, and demonstrate that intensity falls as 1/d² — exactly as predicted by the geometric spreading of an isotropic emitter. The practical also tests a deeper experimental subtlety: the measured distance to the source is not the true distance to the active material, because the radioactive sample sits inside a housing whose physical centre is offset from the housing's outer face. Extracting this "absorbing distance" (or "geometric offset", d₀) is the core data-analysis skill of RP7 and is heavily examined in practical-endorsement (CPAC) assessment. This lesson covers the procedure, the analysis, uncertainty handling, radiation safety, and the synoptic role of inverse-square reasoning across the physics curriculum.
Spec mapping (AQA 7408): This lesson covers the procedural and analytical content of Required Practical 7 — the investigation of the inverse-square law for γ radiation from a sealed source. It includes background-count correction, distance measurement and uncertainty analysis, the linearisation 1/√C versus d, the geometric-offset (absorbing-distance) interpretation, and the radiation-safety protocol (ALARP — As Low As Reasonably Practicable; the three-axis controls of distance, time, and shielding). It also covers the CPAC (Common Practical Assessment Criteria) competencies that the practical addresses. Maps to AQA 7408 section 3.8.1.3 (radioactive decay applications) and the practical-endorsement section 3.16 (Required Practical 7). (Refer to the official AQA specification document for exact wording.)
Synoptic links: (1) The inverse-square law is identical in form to the gravitational field of a point mass (section 3.7.2), the electric field of a point charge (section 3.7.3), and the intensity of any isotropic radiator (including sound and light) — RP7 is a tactile demonstration of a deep geometric principle that recurs throughout physics. (2) The data-linearisation technique (plotting 1/√C against d to obtain a straight line) is the same data-analysis approach used in capacitor-discharge experiments (ln V vs t) and many other A-Level practicals; mastering it for RP7 transfers directly to other practicals. (3) Background-correction discipline learned in RP7 transfers immediately to the time-dependent decay measurement in the radioactive-decay-equations lesson and to many other experimental contexts (cosmic-ray detection, dark-current correction in photodiodes, etc.).
A radioactive source emitting gamma radiation at a constant rate is approximately a point source of isotropic emission. As the photons propagate outwards, they spread over the surface area of a sphere centred on the source. At distance r, the photon flux (number per unit area per unit time) is:
I(r) = P / (4π r²)
where P is the photon emission rate (photons per second). The flux falls as 1/r²: doubling the distance reduces the intensity by a factor of four; tripling reduces by a factor of nine; and so on.
A detector of fixed effective area A intercepts a count rate (after background subtraction) of:
C(r) ∝ A × I(r) ∝ 1/r²
(provided absorption in the intervening air is negligible — a valid approximation for the distances ~5–50 cm used in school laboratories with ~MeV photons).
The 1/r² law assumes the radiation is not absorbed between source and detector. Air absorbs alpha radiation very strongly (range ~3 cm) and beta moderately (range ~30 cm in air for 1 MeV β). Gamma photons of MeV energy have a mean free path in air of hundreds of metres; air absorption over 5–50 cm distances is therefore negligible. RP7 is only meaningful for gamma sources for this reason.
A typical RP7 source is cobalt-60 (γ at 1.17 and 1.33 MeV) or caesium-137 (γ at 0.662 MeV), each typically a sealed 250 kBq source under strict safety supervision.
graph LR
A["Sealed γ source<br/>(Co-60 or Cs-137, ~250 kBq)<br/>in housing"] --> B["Air gap, distance d"]
B --> C["GM tube end-window"]
C --> D["Counter / data logger"]
D --> E["Display: counts per minute"]
F["Long-handle source holder<br/>(tongs, ~30 cm)"] --> A
G["Background measurement<br/>(no source present)"] --> D
style A fill:#e74c3c,color:#fff
style C fill:#3498db,color:#fff
Components.
Background measurement. With the source still in its shielded storage box and as far from the GM tube as practical (≥ 5 m), record the count over a fixed window (typically 10 minutes). Divide by the time to get the background count rate per minute, C_bg. Background varies from day to day with cosmic-ray flux and environmental radon, so this measurement must be done on the day of the experiment, not assumed.
Setup. Clamp the GM tube horizontally. Use long-handle tongs to remove the gamma source from its storage box and clamp it (at the same height as the GM tube end-window) at a distance of about 5 cm. Read off the distance, d, from the outer face of the source housing to the centre of the GM tube end-window, measured to ±1 mm with the metre rule.
First reading. Take a count over a fixed window of typically 60 s (longer at greater distances where the count rate is small). Record the raw count C_raw. Calculate the corrected count rate C = C_raw − C_bg × (window time / 60 s) — i.e. subtract a proportional background.
Vary distance. Move the source through a series of distances — typically 5, 10, 15, 20, 25, 30, 40, 50 cm. At each distance, repeat the count measurement (at greater distances, use a longer window — say 120 s or 180 s — because the count rate is much lower). Always record the count over a measured time, then divide.
Repeat. Take 3 readings at each distance and average to reduce statistical fluctuation.
Stop and shield. When the experiment is complete, return the source to its lead-lined storage box using tongs. Verify the GM tube reads close to background again before leaving.
Data analysis. See next section.
A naive analysis would plot C against 1/d² and look for a straight line through the origin. In practice this doesn't quite work — and the reason is the most important pedagogical lesson of RP7.
The radioactive sample sits inside its housing at some unknown depth behind the outer face. The true distance from the active material to the GM tube end-window is:
d_true = d_measured + d₀
where d₀ is the geometric offset (also called the absorbing distance because part of it represents distance the photon traverses inside the source housing before escaping). For a typical school source, d₀ ≈ 1–3 cm. Similarly, the GM tube's active volume sits slightly behind its end-window, contributing a smaller offset, but this is usually combined into a single effective d₀.
The corrected inverse-square law is therefore:
C ∝ 1/(d_measured + d₀)²
Equivalently:
1/√C ∝ (d_measured + d₀) = d_measured + d₀
So plotting 1/√C against d_measured yields a straight line with:
This linearisation is the standard data-analysis step for RP7 and is rewarded heavily in CPAC and exam mark schemes.
graph LR
A["Measure C at varied d"] --> B["Subtract background"]
B --> C["Compute 1/√C"]
C --> D["Plot 1/√C vs d"]
D --> E["Straight line<br/>gradient k, x-intercept = −d₀"]
E --> F["Verify inverse-square law<br/>(line is straight)"]
E --> G["Extract d₀<br/>(geometric offset)"]
style E fill:#27ae60,color:#fff
Suppose a student records the following corrected count rates (counts/min) at various distances:
| d_measured (cm) | C (corrected, /min) |
|---|---|
| 5 | 1024 |
| 10 | 400 |
| 15 | 196 |
| 20 | 121 |
| 25 | 81 |
| 30 | 57.8 |
| 40 | 33.1 |
| 50 | 21.2 |
Computing 1/√C for each point (with C in counts/min):
| d (cm) | C (/min) | 1/√C |
|---|---|---|
| 5 | 1024 | 0.0313 |
| 10 | 400 | 0.0500 |
| 15 | 196 | 0.0714 |
| 20 | 121 | 0.0909 |
| 25 | 81 | 0.1111 |
| 30 | 57.8 | 0.1316 |
| 40 | 33.1 | 0.1739 |
| 50 | 21.2 | 0.2172 |
Plotting 1/√C (vertical axis) against d (horizontal axis), the data should fall on a straight line. Linear regression gives:
So d₀ ≈ 2.5 cm — the active sample sits about 2.5 cm behind the outer face of the housing (as far as the GM tube is concerned).
The straightness of the line (correlation coefficient r² > 0.99 in this synthetic dataset) confirms the inverse-square law.
If we naively plot C against 1/d² without correcting for d₀:
| d (cm) | C (/min) | 1/d² (cm⁻²) |
|---|---|---|
| 5 | 1024 | 0.0400 |
| 10 | 400 | 0.0100 |
| 15 | 196 | 0.00444 |
| 20 | 121 | 0.00250 |
| 30 | 57.8 | 0.00111 |
If we fit C = K/d², we expect the same K at every d. Computing K = C × d²:
| d (cm) | C × d² (cm² /min) |
|---|---|
| 5 | 25 600 |
| 10 | 40 000 |
| 15 | 44 100 |
| 20 | 48 400 |
| 30 | 52 020 |
K is not constant — it grows as d grows. This is the signature of a non-zero d₀. The "true" K (computed using d_true = d + 2.5 cm) is the same at every distance:
| d_meas (cm) | d_true (cm) | C × d_true² (cm² /min) |
|---|---|---|
| 5 | 7.5 | 57 600 |
| 10 | 12.5 | 62 500 |
| 15 | 17.5 | 60 025 |
| 20 | 22.5 | 61 256 |
| 30 | 32.5 | 61 081 |
Now K ≈ 60 000 cm² /min is approximately constant across the range, confirming both the inverse-square law and the d₀ correction.
Counting (Poisson) statistics. For a count C, the statistical uncertainty is √C (Poisson distribution). Relative uncertainty = 1/√C. At C = 100 counts, relative uncertainty = 10 %; at C = 10 000, relative uncertainty = 1 %. Always take long counts to drive √C-noise down.
Background variability. The background itself fluctuates from minute to minute. Take a longer background measurement (10 min) to reduce its statistical uncertainty.
Distance measurement. With a metre rule, distance can be read to ±1 mm. At d = 5 cm, this is 2 % relative; at d = 50 cm, 0.2 %. Distance dominates uncertainty at short distances; counting dominates at long distances.
GM tube dead time. After each detection event, a GM tube is "dead" for ~100 μs and cannot register another. At very high count rates (> 10 000 /min), this becomes significant. RP7 typically stays well below this threshold.
Source-housing geometry. d₀ is itself uncertain (typically ±0.5 cm), which propagates into the gradient.
For the linearised plot (1/√C vs d), the uncertainty in 1/√C is:
Δ(1/√C) / (1/√C) = (1/2) × ΔC / C = (1/2) × √C / C = 1/(2√C)
So plot 1/√C as the y-value, with vertical error bars of size 1/(2√C). Horizontal error bars are the distance uncertainty Δd ≈ ±1 mm. The slope can be extracted by linear regression with weights inversely proportional to the error-bar size squared.
At d = 50 cm with a 120-second window, the raw count is roughly C_raw = 70 (including background ≈ 20). Corrected count over the window is C ≈ 50 counts → per minute C = 25 counts/min over 2 min → corrected rate ~21 /min.
Uncertainty from Poisson statistics: √70 ≈ 8.4 counts on C_raw (12 % relative), plus √20 ≈ 4.5 counts on background. Combined uncertainty on C ≈ √(70 + 20) ≈ 9.5 counts (19 % relative on corrected total).
To reduce this to 5 %, extend the count window to (19/5)² = 14× = 28 minutes per reading.
Counting uncertainty grows as 1/√(count window) — this trade-off between accuracy and total experiment time is a recurring practical theme.
The principle that doses should be As Low As Reasonably Practicable (ALARP) governs all radiation work. The three axes of dose minimisation are:
| Axis | Action | Why it works |
|---|---|---|
| Distance | Stay as far from the source as possible | Inverse-square law: doubling distance reduces dose by 4× |
| Time | Minimise time near the source | Dose is proportional to exposure duration |
| Shielding | Use lead-lined storage; lead/concrete walls | Gamma intensity decreases exponentially in absorbing material |
A school-laboratory radiation protocol typically includes:
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