Detection limits (MDA / MDC)
Critical level, detection limit and minimum detectable activity for fixed counting, plus scan MDC with observer efficiency.
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Three numbers that are not the same
Detection limits are widely quoted and widely confused, because three distinct quantities all get called "the detection limit". Currie's framework separates them, and keeping them apart is the difference between a defensible measurement report and an argument.
The critical level LC is a decision threshold applied after the measurement. You compare a single net count against it and declare "detected" or "not detected". It is set so that a blank sample exceeds it only with the false-positive rate you chose — 5% at the usual k = 1.645. LC = k√(2B) when the background is counted for the same time as the sample.
The detection limit LD is a property of the method decided before the measurement. It is the true amount that will produce a result above LC with your chosen confidence, so it accounts for false negatives as well: LD = k² + 2LC, the familiar 2.71 + 4.65√B at k = 1.645.
The minimum detectable activity is LD converted into units someone can act on, by dividing by the counting efficiency, the count time, and any yield or aliquot factor. It is the only one of the three that can be compared against a regulatory limit. Reporting LC as "the MDA" understates the method's capability by roughly a factor of two and is the single most common error in this area.
What actually improves an MDA
Because LD grows with the square root of the background counts while the denominator grows linearly with time, MDA improves only as 1/√t. Counting four times as long buys a factor of two. Counting sixteen times as long buys a factor of four. This is why chasing a lower MDA by extending count times reaches a practical wall quickly.
Efficiency, by contrast, enters linearly. Doubling the counting efficiency halves the MDA outright. Moving the detector closer, choosing a better geometry, or using a detector matched to the emission is almost always a better investment than a longer count. Reducing background helps as 1/√B, so shielding a counter is worth roughly as much as counting longer, and usually costs less in throughput.
Scanning is a different problem
A fixed count has a defined observation time. A scan does not: the probe passes over any given spot for only as long as it takes to traverse its own width. That observation interval, the detector width divided by the scan speed, sets the number of background counts available to distinguish a source from noise, and it is usually one or two seconds.
The scan MDC then includes something no instrument specification can supply — the surveyor. The detectability index d′ describes an ideal observer's ability to notice a brief change in signal, and the observer efficiency p accounts for the fact that a real person listening to a rate meter while walking does not achieve it. MARSSIM conventionally takes p = 0.5, which raises the required count rate by √2. Halving the scan speed doubles the observation interval and improves the MDC by √2 — usually the cheapest improvement available, and the reason a documented scan speed belongs in the survey plan.
Surface efficiency
Contamination measurements carry a second efficiency: the fraction of emissions that actually leave the surface rather than being absorbed in it or directed inward. Conventional values are 0.5 for most situations and 0.25 where beta absorption in the substrate is significant. Both efficiencies multiply, so a 20% detector efficiency and a 0.25 surface efficiency give a total of 5% and an MDC twenty times the naive figure.