Poisson counting statistics

The water thickness measurement uncertainty due to neutron counting statistics can be calculated from Poisson counting statistics. For a random process, the standard deviation, Ah in the observed counts I is A/ = y7. The number of neutrons in the incident, or open beam, I0, is the product of the neutron fluence rate (cur2 s 1), integration time T (s), integration area A (cm2), and neutron detection efficiency, ip... 

Poisson counting statistics ultimately limit signal-to-noise ratios in conventional ICP-MS instruments. The improvement achieved by increased intensity is directly proportional to the square root of the counts s . The temptation is, therefore, to work with ever-larger counts. However, the detectors (usually electron multipliers) will have a limit to their linearity and will ultimately saturate. Even over the normal linearity range, typically up to 10 counts s , there will be an influence from the detector s dead time . There is no need to correct for dead time if a matching procedure is being adopted, but in other cases, for very accurate measurements, the multiplier will need to be characterised for dead time. 

Many common mathematical formulas used for calculations of uncertainty, detection capability (see Section 10.4), and quality control in the radioanalytical chemistry laboratory are based on the assumption of Poisson counting statistics and are valid only if that assumption is valid. In particular, the formulas depend on the fact that the mean and variance of a Poisson distribution are numerically equal. For this reason it may be a good idea to test the Poisson assumption from time to time. 

The uncertainty due to varying instrument background can be significant. If the background varies, the assumption of pure Poisson counting statistics to evaluate the uncertainty of the net count rate for a sample may seriously underestimate the uncertainty. Options include replicate background measurements to determine its uncertainty (Type A evaluation), or to evaluate an additional component of uncertainty to be added to the Poisson counting uncertainty. 

In radioanalytical chemistry, the critical value can be calculated by either of two common approaches. One is based on repeated measurements of blank samples, and the other on the assumption of Poisson counting statistics. The former method is generally applicable to any measurement process for which the distribution of measurement results is approximately normal. The latter method is useful when the Poisson assumption is valid, but it may give misleading results in other situations. 

In many cases, the assumption of pure Poisson counting statistics is invalid because other sources of variability affect the distribution of measurement results. In these cases, the laboratory should consider determining the critical value and MDA based on a series of blank samples, as in Eqs. (10.18) and (10.19). When the Poisson model is valid but the background level is so low that the Poisson distribution is not approximately normal, other formulas for the critical value and MDA tend to give the best results, as discussed in MARLAP (EPA 2004). 

This is the ultimate precision attainable and the isotope ratio precision observed in practice should be compared with that predicted by Poisson counting statistics to evaluate whether or not further progress can be made. 

Gray et al. investigated this statement in practice and adapted their quadrupole-based ICP-MS instrument such that as many sources of noise as possible were eliminated, or their contribution minimized [93]. This included the use of a free expansion interface (without skimmer), a bonnet to shield the ICP from the surrounding atmosphere, and the introduction of the analyte (Xe) in gaseous form. Under these extraordinary conditions, they obtained a measurement precision equal to that calculated on the basis of Poisson counting statistics [down to 0.02% relative standard deviation (RSD)]. With a normal (two-cone) interface and sample introduction via pneumatic nebulization, they still obtained extraordinary values of around 0.05%, but at analyte concentrations that are unusually high (100 mg Ag 1 ), an isotope ratio close to 1 ( ° Ag/ ° Ag), and with the sheathing bormet mentioned earlier. 

When focusing attention on single-collector ICP-MS, the measurement precision, usually expressed as the RSD estimated on the basis of N subsequent measurements, typically significantly exceeds that predicted on the basis of Poisson counting statistics. This is a consequence of the combination of the ICP as a rather noisy ion source with sequential monitoring of the intensity of the ion beams of interest. 

Figure 2.23 ° Ag/ Ag isotope ratio precision (RSD for 10 replicate measurements) as a function of the acquisition time per nuclide (a) predicted on the basis of Poisson counting statistics (b) observed experimentally. Reproduced with permission of the Royal Society of...

Figure 2.24 Internal isotope ratio precision (RSD) for ° Ag/ ° Ag as obtained using a quadrupole-based instrument equipped with a collision/reaction cell. Filled squares, vented cell open squares, with Ne as an inert collision gas, introduced into the cell at a flow rate of 2 ml min The dotted line represents the precision as predicted by Poisson counting statistics. Reproduced with permission of the Royal Society of Chemistry from [99].

The first term in this expression represents a correction for the non-normahty of Poisson counting statistics at low total counts [44] and can be neglected if, as generally applies ... 

Fig. 1. Differential K-band counts. The error bars represent only the Poisson counting statistics. Large scale stmcture will increase the errors further.

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