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Measure the half-life of barium-137m with a Geiger counter

Elute a few drops containing ¹³⁷ᵐBa from a ¹³⁷Cs/¹³⁷ᵐBa isotope generator, count them with a Geiger counter interval after interval, subtract the background, plot ln(count rate) against time and get the half-life — with its uncertainty — from the slope.

Difficulty ●●○ ≈ 25 min

Aim Determine the half-life of ¹³⁷ᵐBa and estimate the uncertainty of the result from counting statistics.
Virtual experiment · ¹³⁷Cs/¹³⁷ᵐBa isotope generator + Geiger counter

Measuring the half-life of barium-137m

00000Last intervalClock 0 s¹³⁷ᵐBa

① Measure the background

② Elute the generator and start the clock

③ Count, interval after interval

050100150200250300Interval midpoint t / s−2−1012345ln(net rate / s⁻¹)

At least 3 intervals with positive net counts are needed for a fit.

Counts are Poisson draws (deterministic pseudo-random numbers, reproducible per sample number): each interval’s expected count is the net rate R₀·2^(−t/t½) integrated over the interval, plus background. R₀ and the background vary with the sample number within the range typical of teaching labs. The fit is weighted least squares on ln R, with weights from counting statistics, u(n) = √(gross counts + variance of the background correction). Dead time, changes in geometry and ¹³⁷Cs breakthrough are not modelled.

Procedure

  1. Before any sample is present, count the background for 300 s and find the background rate and its uncertainty √N/t.
  2. Elute the generator (t = 0) and put the dish of eluate in a fixed position under the counter window.
  3. Choose a counting interval (say 20 s) and count interval after interval, recording at least 10 of them (about four half-lives).
  4. For each interval subtract the background, work out the net count rate and its uncertainty, and plot ln(net rate) against the midpoint of the interval.
  5. Fit a weighted straight line to get the slope −λ, then t½ = ln 2/λ; finally reveal the accepted value and see how many standard uncertainties you are off.

What to notice

  • Every count fluctuates: a count N has a standard deviation of about √N. Early counts are large with small relative errors; late counts are small and their error bars grow.
  • ln(net rate) against time is a straight line — the signature of first-order decay; the slope does not depend on the starting count rate.
  • The last intervals are nearly all background, and after subtraction may even go negative and must be dropped: longer counting helps, but not once the source has faded into the background.
  • Repeat with a new sample: the results agree within their uncertainties. Changing the interval from 10 s to 30 s hardly changes the fitted uncertainty.

In a real lab

  • The ¹³⁷Cs/¹³⁷ᵐBa generator is a sealed teaching source; the eluate holds ¹³⁷ᵐBa and a trace of ¹³⁷Cs. Wear gloves, work over a tray, and collect the waste according to your school’s rules for radioactive sources.
  • At high count rates the Geiger tube’s dead time (about 100 µs) loses counts and makes early rates too low; increase the distance to the tube or apply a dead-time correction.
  • Keep the sample in exactly the same position throughout: any change in geometry changes the count rate for reasons that have nothing to do with decay.
  • ¹³⁷ᵐBa emits 662 keV γ rays, which a Geiger tube detects with an efficiency of only a few per cent, so the count rate is far below the sample’s actual activity.

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