Introdcution to Detection Limits

Detection limit is commonly understood to be the smallest concentration we can measure with a particular technique. In fact it is the point at which we can make a decision whether the element or compound is present or not. To be able to measure it we need at least three times the detection limit. Three times the detection limit is often called the limit of determination. A different term used for detection limits is "Limits of Detection" with the abbreviation LoD.

By convention and from statistics, detection limit (DL) is defined as the concentration corresponding to a signal three times the noise level of the background.

Measuring DLs

There are several ways to measure detection limits. A quick (rough) way is simply to divide the BEC (background equivalent concentration, i.e. the concentration intercept on the calibration curve) by a number, people usually use 50 or 30. The number depends on the typical noise level in the instrument. This is explained further below.

Another way is to determine the uncertainty on the BEC. A third way is called the signal-to-noise ratio (SNR) method. A favoured approach is the SBR-RSDB method.

SNR Method

The SNR method can be expressed as:

the sensitivity (the slope of the calibration curve of intensity versus composition), where x_A is the net analyte signal (i.e. signal above background) and c₀ the composition of the element in the sample.

Clearly with this method, the detection limit is largely determined by the background signal: its size and its noise level, expressed as RSDB. And the sensitivity of the technique, expressed as the slope of the calibration curve.

SBR-RSDB Method

The SBR-RSDB method can be expressed in two equivalent ways:

where > c is the mass % of the element in the sample being measured, BEC is expressed as mass %, and SBR is the signal-to-background ratio.

Again, the detection limit is largely determined by the background signal: its size and its noise level, expressed as RSDB. And the sensitivity of the technique, expressed as SBR.

We can see that if RSDB is 0.7%, then DL is approximately BEC/50 and the limit of determination is 3xBEC/50. If RSDB is 1%, then DL is about 30, and if RSDB is 2%, then DL is BEC/17.>[Note: if the RSDB is 1%, which is often assumed, then BEC/50 corresponds to two times the noise level on the background and so the limit of determination is taken to be five times the DL, i.e. BEC/10, rather than three times in the more formal approach.]

Detection limits with more formulars

Since, in optical emission spectroscopy, we measure intensities to determine amounts, the detection limit corresponds to the smallest intensity from the analyte that can be measured and distinguished from the background. One method to determine the DL is to measure signals with and without a tiny amount of the analyte. The signal without the analyte is called a “blank”. An alternative to using a blank is to measure the signal in the background at a n close to the emission line of interest. Thus we have two means, one measured with the analyte
 and one without it.
If the true means are m₁ and m₂, then the difference between them is given by

where n₁ and n₂ are the number of measurements for each and s_e is the combined estimate of the standard deviation, given by

where s₁ and s₂ are the standard deviations of the two sets of measurements.

To be sure of having a real signal from the analyte

The detection limit therefore corresponds to

If we make an equal number of measurements with and without the analyte signal, then n = n₁ = n₂, where n is the number of measurements of the backgound or analyte; and if the analyte signal at the detection limit is small compared with the background or blank signal, then where s is the standard deviation of either the background or (background plus analyte) signal. Thus the detection limit simplies to

If n is large (≥ 15), for a 95% confidence, t = 2.0 and √2.t = 2.8, usually approximated to 3. Hence<
Immediately we notice that the detection limit depends on the standard deviation of the background, i.e. on the noise in the background signal, and not on the size of the background signal, though the higher the background signal often the higher the noise. Also we notice that we can reduce the detection limit by taking more measurements, though, as it depends on √n, it is a matter of diminishing returns.

 

Further reading: P W J M Boumans, in R Payling, D G Jones and A Bengtson (Eds), Glow Discharge Optical Emission Spectrometry, John Wiley & Sons, Chichester (1997), pp 440-451.
Th.Nelis and R. Payling, RSC Analytical Spectroscopy Monographs, Glow Discharge Optical Emission Spectroscopy: A practical guide., RSC CambrigdeUK, (2004) p 111.

First published on the web: 15 May 2000.

Author: Richard Payling