Answer to Question #13340 Submitted to "Ask the Experts"

Category: Instrumentation and Measurements

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Q

How do we measure or calculate "error" in gamma spectrometric sample results?

A

As you are aware, the estimation of counting uncertainties associated with gamma spectrometry is a bit more complicated than for some other types of laboratory counting of particulate radiation. The basic principles involved are the same, namely propagation of errors associated with manipulation of counting data, in particular the subtraction of a background counts from gross counts to obtain net counts and the uncertainty associated with these counts.

The major difference in how the uncertainty determination proceeds results from the fact that the background in gamma spectrometry may be variable, with possible contributions coming both from the ambient background radiation, independent of the sample being analyzed, and also from the sample itself as Compton scattered photons from photon energies higher than the photons of interest may produce counts in the region of interest (ROI) associated with the photopeak being analyzed. In most instances the ambient background does not exhibit a photopeak in the ROI, and we shall assume that is the case for in this discussion. In such cases, the background under the photopeak is typically determined by drawing a curve, most commonly a straight line connecting a few points on each side of the photopeak. Generally equal numbers of points are chosen on each side of the photopeak. When a straight line is envisioned connecting these points, if N channels have been selected on each side of the photopeak, CN is the combined number of counts in those 2N channels, and P is the number of channels in the photopeak ROI, then the expected number of background counts, CB, in the ROI is P(CN/2N). If CG is the number of gross counts in the ROI, the number of net counts in the ROI is then

Cnet = CG – CB = CG – PCN/2N.  (1)

If both CG and PCN/2N are treated as integral counts, each with a Poisson one sigma uncertainty given by the square root of the respective count, the one sigma Poisson uncertainty in Cnet is calculated as

σnet = (CG + PCN/2N)0.5.   (2)

This is based on the fact that when two counts are added or subtracted, the propagated variance in the result is the sum of the variances of the two counts; in Poisson statistics the variance of a count is the count itself, and the standard deviation (one sigma uncertainty) is the square root of the variance.

It is common for much of the current commercial software available for gamma analysis to carry out the uncertainty calculations on a routine basis.

As a brief example, if we observed a photopeak that contained 2000 gross counts in the ROI, which contained seven channels, and we used three channels on either side of the ROI to define the background line, and these six channels contained a total of 540 counts, the net count rate would be

Cnet = 2000 – 7(540/6) = 1370 counts.

The standard deviation in this value would be

σnet = (2000 + 7(540/6))0.5 = 51 counts.

If one wanted results in terms of count rate, a similar rationale would apply, simply recognizing that, neglecting decay during the counting interval, the net count rate is the net count divided by the counting time, and the variance in a single count rate is given by the count rate divided by the counting time. Using notation above, the propagated standard deviation in the net count rate is then

σnet rate = (CG/t2 + PCN/2Nt2)0.5,    (3)

which is also familiarly written using rate notation as

σnet rate = (RG/t + RB/t)0.5.   (4)

For this situation the sample is counted for a fixed amount of time, and this same time applies to both the sample ROI counts of concern and the background counts. If, in the earlier example, the sample had been counted for 100 minutes the propagated one sigma uncertainty in the net count rate of 13.7 cpm would have been +0.51 cpm.

We shall not deal in detail with the case for which the ambient background contains a photopeak in the ROI. In such a situation there would be another background component that would have to be subtracted from the gross ROI count to account for the background photopeak counts in the ROI. This contribution is handled in a mathematically similar way except that the ambient background would have to be evaluated separately to evaluate the contribution in the ROI and adjustments made, when the sample is counted, to account for any difference between the sample counting time and the counting time used to evaluate the ROI peak contribution from the ambient background.

I hope this has been helpful.

George Chabot, CHP, PhD

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