X-ray emission and detection is a random process governed by Poisson counting statistics, and s, a sample estimate of the true population standard deviation, is given by:

where N is the total number of x-ray counts measured on a peak or background.

For count rates (counts per second), the standard deviation s_R becomes:

where T is the count time and R the count rate (= N/T). (Microscan software always prints x-ray counts as dead time corrected count rates.)
 

2s_R is the error value printed out by the Microscan software, but is not strictly correct for net intensity (peak minus background). The correct estimate is given by:

where R_p is count rate on the peak, over time period T_p, and R_b is count rate on the background over time period T_b. As the ratio R_p/R_b increases, the effect of R_b ons_D decreases, so that s_D approximately equals s_R if background is low and peak to background is high.

Although that estimates the error in subtracting background to obtain net peak count rates, in microprobe analysis the further step is taken of ratioing two net peak count rates (on the specimen and the standard).

Again, errors add in quadrature:

Relative standard deviation for a single net peak measurement:

n.b. in Microscan standards tables, parameter T  is the equivalent of T_p , and TB the equivalent of T_b (unless set to TB0.00, in which case, T  equals T_b also).

and for specimen/standard peak ratio:

  Counting Strategy

It can be shown that if R_p is the gross (peak + background) count rate, and R_b the background count rate, then for minimum error in net peak measurements, the count times between peak (T_p ) and background (T_b) should be apportioned by:

total time available to make the measurement equals T_p + T_b

In practice, for a generalised analytical procedure applicable to a variety of compositions, full advantage is not usually taken of this possibility, but it is worth bearing in mind for detailed work on a restricted range of compositions.
 

Detection limits

In order to be able to measure an X-ray peak, we have to have reasonable statistical certainty that we can see it above background. Since the peak count rate and background count rate will be very similar, and peak and background count times should be the same, the formula given above:

 can be reduced to:

and lower limit of detection is often taken as 3s_Rb, i.e. 3 times the standard deviation of the background count rate. This takes the assumption, based on normal distribution statistics, that a signal at a level 3s_Rb above the mean background has a 99.87% probability of not belonging to the background distribution. It is important to note that quantitative analysis is not possible at this signal level, and in fact it represents the limit of qualitative analysis. The limit of determination is the smallest signal that can be quantitatively measured using 3s confidence criteria, and is taken as being 6s_Rb :

or in terms of weight %:

where the figure of merit, m = count rate per % of element of interest (any difference in beam current between specimen and standard measurement should be taken into account).

JEOL JXA-8800 software appears to use s_Rb as the measure for lld, which is unduly optimistic.

For Microscan standard tables, m = STR / CON

Sources and further reading:

S.J.B. Reed (1996) Electron Microprobe Analysis and Scanning Electron Microscopy in Geology, Cambridge University Press. (A good introductory text, aimed at users with little or no previous knowledge of the subject.)

S.J.B. Reed (1993) Electron Microprobe Analysis, Cambridge University Press. (A more advanced text, covering the subject in depth, with detailed sections on Physics of x-ray generation and correction procedures)

V.D. Scott, G. Love & S.J.B. Reed (1995) Quantitative Electron-probe Microanalysis, Ellis Horwood. (An advanced text which covers correction procedures in great detail, but has useful general introductory material)

P.J. Potts et al., eds. (1995)  Microprobe Techniques in the Earth Sciences,  Min. Soc. Series 6, Chapman & Hall. (A number of techniques presented by specialist contributors, aimed at providing general introductions; includes EPMA, PIXE, SIMS, Laser ICPMS)

P.J. Potts (1987) A Handbook of Silicate Rock Analysis, Blackie. (Comprehensive and detailed, covering a wide range of techniques, but approachable at both elementary and advanced levels)