A Six-Hour Source, Background Counts and an Open Day
A school physics department keeps a sealed source whose radionuclide has a half-life of 6.0 hours. At 09:00 a student records 480 counts in 60 s with the source in its holder in front of a GM tube. She removes the source and records 60 counts in 60 s. At 15:00 the same group repeats the measurement and records 265 counts in 60 s, having checked the background again and obtained 60 counts in 60 s. The teacher says the result 'proves' the half-life is exactly 6.0 hours. One student disagrees, arguing that the second reading is too low and that the source must be a mixture of two radionuclides. The department also plans to use the same source for open-day demonstrations six months later.
1. Correct both readings for background, compare with the expected value, and say whether the result contradicts a half-life of 6.0 hours.
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Subtract the background from each gross count:
- 09:00: 480 - 60 = 420 counts per minute
- 15:00: 265 - 60 = 205 counts per minute
6.0 hours is exactly one half-life, so the expected 15:00 value is 420/2 = 210 counts per minute, i.e. a gross count of 270.
Radioactive decay is random, so a count of about 205 carries a standard uncertainty of roughly the square root of 205, about 14 counts (7%). The difference between 205 and 210 is 5 counts, about 0.35 of a standard deviation - well within normal fluctuation.
Conclusion: the data are consistent with a 6.0 hour half-life. There is no evidence for a mixture of two nuclides; a mixture would need contrived proportions and would show a curve that is not a simple exponential over several readings. The teacher's word 'proves' is too strong - one pair of readings can only support the value.
2. What half-life would the students deduce if they forgot to subtract the background?
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Using the gross counts, the ratio is 480 / 265 = 1.81, not 2.
Since 2^(t/T) = 1.81 with t = 6.0 h:
t/T = log 1.81 / log 2 = 0.593 / 0.693 = 0.856
T = 6.0 / 0.856 = 7.0 hours
So omitting the background gives a half-life about 17% too long. The reason is that the background does not decay, so it makes up an increasing fraction of the count as the source weakens - here 60 out of 265, about 23%. The error grows with time and would eventually make the source appear never to decay.
The correct procedure is to measure background with the source removed, in the same room and with the same tube, subtract it from every reading, and repeat counts to reduce statistical noise.
3. Advise the department on using this source six months later, and on how reliable a half-life measured over one half-life is.
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Six months later: 180 days is about 4.3 x 10^3 hours, which is 4.3 x 10^3 / 6.0 = about 720 half-lives. The activity falls by a factor of 2^720, which is zero for any practical purpose - the source will give no measurable count above background. It cannot be used for a demonstration. If a source is wanted for years, a long-half-life nuclide (tens or hundreds of years) is appropriate, or a generator that produces a short-lived nuclide on site.
Reliability of the measurement: two readings one half-life apart give only a rough check. Better practice is to take readings every hour over at least three or four half-lives, subtract the background, and plot log(activity) against time; a straight line confirms exponential decay, and its gradient gives the half-life. Each point should carry an uncertainty of about the square root of the count. Also check the count time: at 6 hours the activity falls by only about 0.3% during a 60 s count, so that is negligible, but for a very short-lived source it would not be.