Use measurements of background radiation to determine a corrected count rate

5.2.1 Detection of Radioactivity – Corrected Count Rate

Background Radiation

Background radiation is the natural ionising radiation that is present everywhere in the environment. It originates from cosmic rays, terrestrial radionuclides (e.g. potassium‑40, radon), and man‑made sources. When measuring a radioactive sample, the detector records both the sample’s radiation and the background. To obtain the true activity of the sample, the background contribution must be subtracted.

Measuring Count Rates

A Geiger–Müller counter (or other radiation detector) records the number of ionising events, \(N\), during a counting time, \(t\). The raw count rate is

\$ R{\text{raw}} = \frac{N{\text{raw}}}{t} \quad \text{(counts s}^{-1}\text{)} \$

Similarly, the background count rate is

\$ R{\text{bg}} = \frac{N{\text{bg}}}{t_{\text{bg}}} \$

It is best practice to use the same counting time for both measurements, but if the times differ, the rates must be calculated separately before subtraction.

Corrected Count Rate

The corrected count rate, representing the sample’s contribution only, is obtained by subtracting the background rate from the raw rate:

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

In terms of counts:

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

Example Calculation

Suppose a Geiger counter records:

• Raw counts: \(N_{\text{raw}} = 1200\) in \(t = 10\) s
• Background counts: \(N{\text{bg}} = 200\) in \(t{\text{bg}} = 10\) s

Compute the rates:

\$ R_{\text{raw}} = \frac{1200}{10} = 120 \; \text{counts s}^{-1} \$

\$ R_{\text{bg}} = \frac{200}{10} = 20 \; \text{counts s}^{-1} \$

Corrected rate:

\$ R_{\text{corr}} = 120 - 20 = 100 \; \text{counts s}^{-1} \$

Corrected counts:

\$ N_{\text{corr}} = 1200 - 200 = 1000 \$

Uncertainty in Corrected Count Rate

Counts follow Poisson statistics, so the standard uncertainty in a count is \( \sigma_N = \sqrt{N} \). The uncertainty in a rate is therefore

\$ \sigma_R = \frac{\sqrt{N}}{t} \$

When subtracting two independent rates, the uncertainties combine in quadrature:

\$ \sigma{R{\text{corr}}} = \sqrt{ \sigma{R{\text{raw}}}^2 + \sigma{R{\text{bg}}}^2 } \$

For the example:

\$ \sigma{R{\text{raw}}} = \frac{\sqrt{1200}}{10} \approx 3.46 \; \text{counts s}^{-1} \$

\$ \sigma{R{\text{bg}}} = \frac{\sqrt{200}}{10} \approx 1.41 \; \text{counts s}^{-1} \$

\$ \sigma{R{\text{corr}}} = \sqrt{3.46^2 + 1.41^2} \approx 3.78 \; \text{counts s}^{-1} \$

Thus the corrected count rate is \(100 \pm 3.8\) counts s\(^{-1}\).

Practical Tips

• Use the same counting time for sample and background measurements.
• Record background counts before and after the sample to detect any drift.
• If the counting times differ, calculate rates first and then subtract.
• Always report the corrected rate with its uncertainty.
• Check that the background is stable; large variations indicate detector or environmental issues.

Summary

To determine the true activity of a radioactive sample:

1. Measure the raw counts and the background counts over the same time interval.
2. Calculate the raw and background count rates.
3. Subtract the background rate from the raw rate to obtain the corrected rate.
4. Propagate uncertainties using Poisson statistics and quadrature addition.

Accurate background subtraction is essential for reliable measurements in IGCSE Physics 0625.

Data Table