Use measurements of background radiation to determine a corrected count rate

5.2.1 Detection of Radioactivity – Corrected Count Rate

1. Background Radiation

• Definition: Natural ionising radiation that is present everywhere in the environment.
• Typical sources (Cambridge syllabus)
  • Cosmic rays – high‑energy particles from outer space.
  • Terrestrial radionuclides – e.g. potassium‑40, uranium‑series, thorium‑series.
  • Man‑made sources – radon gas in houses, fallout from nuclear tests.
• Why subtract it? The detector records both the sample’s radiation and this constant background. Subtracting the background removes the offset and isolates the sample’s true activity.

2. Detectors Used in the IGCSE Laboratory

• Geiger–Müller (GM) tube – “click” counter, works for α, β and γ radiation.
• Scintillation counter – crystal + photomultiplier, higher efficiency for γ‑rays.
• Proportional counter – gives pulse height proportional to energy; useful for β and low‑energy γ.
• Solid‑state detector (silicon or germanium) – fast response, good energy resolution, mainly for γ.

All detectors are connected to a counting unit (digital counter or computer interface) that records the number of ionising events, N, in a chosen counting time, t.

3. Key Symbols & Units

4. Measuring Count Rates

1. Raw (sample + background) count rate

$$ R_{\text{raw}} = \frac{N_{\text{raw}}}{t_{\text{raw}}}\quad\text{(cps or cpm)} $$

2. Background count rate (measured separately)

$$ R_{\text{bg}} = \frac{N_{\text{bg}}}{t_{\text{bg}}} $$

Best practice: Use the same counting time for both measurements (normally 30 s or 1 min). If the times differ, calculate the two rates first and then subtract.

5. Corrected Count Rate

Subtract the background contribution:

$$ R_{\text{corr}} = R_{\text{raw}} - R_{\text{bg}} $$

When the counting times are identical, you can work directly with the raw counts:

$$ N_{\text{corr}} = N_{\text{raw}} - N_{\text{bg}} $$

5.1 Example – Same Counting Time (30 s)

Measurement	Counts N	Time t (s)	Rate R (cps)
Raw sample	1 200	30	40.0
Background	200	30	6.7
Corrected	1 000	30	33.3

5.2 Example – Different Counting Times

Suppose the background is measured for 20 s while the sample is measured for 30 s.

$$ R_{\text{bg}} = \frac{200}{20}=10\;\text{cps} $$ $$ R_{\text{raw}} = \frac{1\,200}{30}=40\;\text{cps} $$ $$ R_{\text{corr}} = 40 - 10 = 30\;\text{cps} $$

6. Propagation of Uncertainty (Poisson Statistics)

• Counts follow Poisson statistics → standard uncertainty in a count:
$$ \sigma_N = \sqrt{N} $$
• Uncertainty in a rate (same counting time for numerator and denominator):
$$ \sigma_R = \frac{\sqrt{N}}{t} $$
• When two independent rates are subtracted, uncertainties add in quadrature:
$$ \sigma_{R_{\text{corr}}}= \sqrt{\sigma_{R_{\text{raw}}}^{2}+\sigma_{R_{\text{bg}}}^{2}} $$

Numerical example (same 30 s counting time)

$$ \sigma_{R_{\text{raw}}}= \frac{\sqrt{1\,200}}{30}= \frac{34.64}{30}=1.15\;\text{cps} $$ $$ \sigma_{R_{\text{bg}}}= \frac{\sqrt{200}}{30}= \frac{14.14}{30}=0.47\;\text{cps} $$ $$ \sigma_{R_{\text{corr}}}= \sqrt{1.15^{2}+0.47^{2}}=1.24\;\text{cps} $$

Result: R_corr = 33.3 ± 1.2 cps.

7. From Corrected Count Rate to Activity (Becquerels)

The activity, A, of the sample is related to the corrected count rate by the detector efficiency, ε:

$$ A = \frac{R_{\text{corr}}}{\varepsilon}\qquad\text{(Bq)} $$

Important note: ε is specific to the detector‑radiation combination (e.g. a GM tube may have ε ≈ 0.30 for β‑particles from a thin source but ε ≈ 0.05 for γ‑rays). Always use the efficiency value supplied by the teacher or obtained from a calibrated source.

Worked example

• Corrected count rate: 33.3 cps
• Detector efficiency for the radiation type used: ε = 0.25 (25 %)

$$ A = \frac{33.3}{0.25}=133.2\;\text{Bq} $$

Uncertainty (propagating only the rate uncertainty):

$$ \sigma_A = \frac{\sigma_{R_{\text{corr}}}}{\varepsilon}= \frac{1.24}{0.25}=4.96\;\text{Bq} $$

Hence, A = 133 ± 5 Bq (rounded to 2 s.f.).

8. Background‑Subtraction Table Template (exam‑ready)

Sample	Background	Corrected
______ cpm	______ cpm	______ cpm

Insert your measured values in the blanks; the “Corrected” column is simply “Sample – Background”. This format is commonly expected in Cambridge exam answers.

9. Command‑Word Reminder

The syllabus often uses the command words calculate and explain for this topic. In your answer:

• Calculate – show the numerical steps for raw rate, background rate, corrected rate, uncertainty, and (if required) activity.
• Explain – describe why background must be subtracted, why uncertainties are added in quadrature, and the role of detector efficiency.

10. Practical Tips (Box)

11. Summary Checklist

1. List the three sources of background radiation.
2. Measure raw counts and background counts (same counting time preferred).
3. Calculate raw and background count rates (cps or cpm).
4. Subtract to obtain the corrected count rate (use the three‑column table if required).
5. Propagate uncertainties using Poisson statistics and quadrature addition.
6. If asked, convert the corrected rate to activity using the appropriate detector efficiency.
7. Report the final value with its uncertainty, correct unit and the appropriate command‑word wording.

12. Suggested Diagram