Half-Life Counting with Background
A student measures the activity of a short-lived source with a Geiger counter. Background is measured for 10 minutes: 240 counts. The source is then counted for 1-minute intervals: 820, 610, 455, 340, 255, 190. The student plots raw counts versus time, fits an exponential, and reports a half-life of 1.4 min. A classmate says the background is negligible because it is only 24 counts/min compared with 820, and that the first interval should be discarded as 'settling time.'
1. Correct the data for background and recompute the half-life from the first and second intervals.
Read the worked answer
Background rate = 240/10 = 24 counts/min. Subtract from each interval:
- 820 - 24 = 796
- 610 - 24 = 586
- 455 - 24 = 431
- 340 - 24 = 316
- 255 - 24 = 231
- 190 - 24 = 166
Using intervals 1 and 2: ratio = 586/796 = 0.736. Half-life = ln 2 / ln(1/0.736) * 1 min = 0.693/0.306 = 2.26 min. So the corrected half-life is about 2.3 min, not 1.4 min. The uncorrected fit is biased short because background inflates the late-time counts disproportionately.
2. Evaluate the classmate's two claims.
Read the worked answer
Claim 1 (background negligible): Wrong. Background is 24/820 ≈ 3% of the first interval but 24/190 ≈ 13% of the last, and it grows relatively as the source decays. It systematically flattens the decay curve and biases the half-life. Background must be subtracted, especially for late intervals.
Claim 2 (discard first interval): Not justified as 'settling time' without evidence. The first interval is the most statistically precise (largest counts). Discarding it throws away the best data. If there is a real dead-time or settling effect, the student should demonstrate it with a repeat measurement, not assume it.
3. The student wants to improve the half-life estimate. Propose a better analysis and state one check.
Read the worked answer
Fit all background-subtracted points on a log-linear plot (ln N vs t), where exponential decay is a straight line; the slope gives the decay constant. Use a weighted least-squares fit with Poisson weights (weight ∝ N) so high-count points dominate appropriately. Alternatively, count longer per interval to reduce statistical noise, and repeat the background measurement.
A good check is a residual plot: if the fit is good, residuals should scatter randomly about zero. A systematic curvature would indicate a second component (two isotopes) or dead-time losses at high count rates. The student should also quote the statistical uncertainty on the half-life from the fit.