How Long Must the Source Be Stored?
A school stores a sealed gamma source whose half-life is 6.0 hours. With the source in its holder, a Geiger–Müller tube records 320 counts in 60 s. With the source removed, the same tube records 40 counts in 60 s in the same room. The radiation officer requires the count rate above background to fall below 10% of its present value before the source is moved to a general store. A technician suggests counting for 10 s instead of 60 s 'because the rate is high enough'. A teacher predicts the source will be safe after 'about two half-lives'. The tube's dead time is negligible at these rates.
1. Calculate the storage time needed. Evaluate the teacher's prediction.
Read the worked answer
Background = 40 counts/min; source plus background = 320 counts/min; net rate = 280 counts/min.
Target net rate = 10% of 280 = 28 counts/min.
280 → 28 is a factor of 10, so the number of half-lives is log₂10 = 3.32.
Time = 3.32 × 6.0 h = 19.9 h, about 20 hours.
Two half-lives would reduce 280 to 70 counts/min, which is 2.5 times the target - not safe. The teacher's rule of thumb is roughly right for 'much reduced' but wrong for a specific 10% criterion, because the required number of half-lives depends on the factor chosen, not on the source itself.
2. The technician counts for 10 s instead of 60 s. Assess whether that is acceptable, particularly near the decision point.
Read the worked answer
In 10 s the source-plus-background count is about 53 and the background about 7, giving a net of about 47 counts. The random uncertainty is roughly √(53 + 7) ≈ 7.7 counts, i.e. about 17%. That is tolerable for simply confirming a source is present.
At the decision point the net rate is 28 counts/min, which is only about 4.7 counts in 10 s, with σ ≈ √(4.7 + 0.7) ≈ 2.3 counts - about 50%. The reading could not distinguish 'safe' from 'not yet safe'. Near the threshold a much longer count, of several hundred seconds, is needed. Note that uncertainty falls only as √t, so lengthening the count from 10 s to 100 s reduces σ by about a factor of 3, not 10.
3. State the assumptions built into the 20-hour answer and suggest a practical procedure that does not rely on a single calculation.
Read the worked answer
Assumptions: background stays at 40 counts/min (it can drift with weather, radon, or other sources); dead time is negligible; the stored source is the only extra activity; and '10% of present value' means the net rate, not the meter reading. The 60 s counts themselves carry about 6% uncertainty on the net rate, so 20 h is a best estimate with perhaps ±1 h of margin.
A sounder procedure is to store the source behind lead, then re-measure at about 18 h and again at 22 h with long counts, using the same geometry and a background check each time. Decide from those measurements rather than from one arithmetic prediction. Record the dates and readings so the decay curve itself can be checked against the stated half-life.