What it does
Radioactive decay is random, so every count has an uncertainty: for N counts the standard deviation is √N. This page does the everyday sums of a counting lab with that in mind.
- Net counts and detection limit. From gross counts (sample plus background) and a separate background count, it gives the net rate with its uncertainty, √(G/t_g² + B/t_b²), and Currie’s two limits. The decision threshold L_C is the net count above which you say “detected”, with a false-positive rate α. The detection limit L_D is the true net count you would detect with probability 1 − β. With a counting efficiency it also gives the activity and the minimum detectable activity (MDA). Exact Poisson limits are shown for the gross and background counts.
- Dead-time correction. A detector is blind for a short time τ after each event. The page corrects a measured rate with the non-paralysable model, n = m / (1 − mτ), and the paralysable one, m = n e^(−nτ) (solved with the Lambert W function), and plots both.
- Counting-time planning. For expected gross and background rates, the best split of a fixed time is t_sample / t_background = √(R_gross / R_background); the page gives that split, the uncertainty it reaches, and the total time for the uncertainty you want.
How to use it
Type the counts and times (in seconds, minutes or hours; the same unit for both), and optionally the efficiency (counts per particle emitted) and the emission yield (particles per decay). Pick α and β (5 % each is usual) and the confidence for the intervals. The other two cards work on their own.
Worked example
A made-up sample gives 1,500 counts in 600 s; the background gives 900 counts in 1,800 s. The net rate is 2 ± 0.067 per s, 3.33 %. With no activity, the net count in 600 s would scatter with σ₀ = 20 counts, so L_C = 32.9 and L_D = 68.5 net counts; the 1,200 net counts are far above L_C, so the page says detected. At 25 % efficiency the activity is 8 ± 0.27 Bq and the MDA is 0.4567 Bq (27.4 dpm).
A counter reading 20,000 per s with a 5 µs dead time is busy 10 % of the time: the true rate is 22,222 per s (non-paralysable) or 22,367 per s (paralysable). With rates of 2.5 and 0.5 per s and 2,400 s in all, count the sample for 1,658 s and the background for 741.6 s to reach 2.34 %; reaching 1 % needs 13,090 s.
Tips
- Count the background for at least as long as the sample when the sample is weak; the split rule tells you how much longer.
- L_C is for deciding about this measurement; L_D (and the MDA) describe what the method can detect, before you measure.
- Keep dead-time losses below about 10 % when you can; the two models start to disagree above that.
FAQ
Why do two “detected” numbers exist? L_C controls false alarms; L_D controls missed detections. A true signal exactly at L_C would be missed half the time, so L_D is about twice as large.
Is the normal approximation good enough? For a few tens of counts or more, yes. For very small counts the exact Poisson limits shown for the gross and background counts are safer.
Is this enough for a regulatory report? No. It covers Poisson statistics only. It is for study and planning, not a substitute for professional judgement or your laboratory’s validated procedures.
Where the numbers come from
- Currie, L. A. (1968), Limits for qualitative detection and quantitative determination, Analytical Chemistry 40, 586 to 593 (decision threshold and detection limit formulas).
- Poisson counting, dead-time models and optimum counting-time division: standard results, for example Knoll, G. F., Radiation Detection and Measurement. Exact Poisson limits: Garwood, F. (1936), Biometrika 28, 437.
- Checked against SciPy (normal and chi-square quantiles, Lambert W, root finding) and against Currie's paired-blank forms L_C = 2.33 √B and L_D = 2.71 + 4.65 √B, in tests written for this page.
- The example counts are made up.
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Last updated 2026-11-09.