Counting statistics

For each absorbed x-ray photon or neutron, the proportional or scintillation counter produces a discrete electric pulse. The flux J of the beam of x-rays or neutrons is measured as the number of counts of such pulses observed per second. If measurements are made repeatedly with a beam of constant flux, the number of counts observed during a fixed time period is not exactly the same, but is rather subject to statistical fluctuations. The arrival time of any one particle (x-ray photon or neutron) is totally uncorrelated with the arrival time of the next particle. The flux J of the particles, 

Minimization of x et with respect to/ shows that the optimum value of/ = fgross/ftotai is given by 

for example, when Jbkgd and Jgr0ss are in the ratio of 1 to 4, the measurement times Jbkgd and Jgr0Ss should be apportioned in the ratio of 1 to 2. 

This derivation ignores the possibility of more than two pulses piling up and is therefore applicable only when the counting losses are not too high. 

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The main sources of error which define the accuracy are counting statistics in tracer concentration measurements, the dispersion of the tracer cloud in the flare gas stream, and the stationarity of the flow during measurements. 

The concentration of is determined by measurement of the specific P-activity. Usually, the carbon from the sample is converted into a gas, eg, carbon dioxide, methane, or acetylene, and introduced into a gas-proportional counter. Alternatively, Hquid-scintiHation counting is used after a benzene synthesis. The limit of the technique, ca 50,000 yr, is determined largely by the signal to background ratio and counting statistics. 

Type of Data In general, statistics deals with two types of data counts and measurements. Counts represent the number of discrete outcomes, such as the number of defective parts in a shipment, the number of lost-time accidents, and so forth. Measurement data are treated as a continuum. For example, the tensile strength of a synthetic yarn theoretically could be measured to any degree of precision. A subtle aspect associated with count and measurement data is that some types of count data can be dealt with through the application of techniques which have been developed for measurement data alone. This abihty is due to the fact that some simphfied measurement statistics sei ve as an excellent approximation for the more tedious count statistics. 

Because X-ray counting rates are relatively low, it typically requires 100 seconds or more to accumulate adequate counting statistics for a quantitative analysis. As a result, the usual strategy in applying electron probe microanalysis is to make quantitative measurements at a limited collection of points. Specific analysis locations are selected with the aid of a rapid imaging technique, such as an SEM image prepared with backscattered electrons, which are sensitive to compositional variations, or with the associated optical microscope. 

The applicability of the Poisson distribution to counting statistics can be proved directly that is, without reference to binomial theorem or Gaussian distribution. See J. L. Doob, Stochastic Processes, page 398. The standard deviation of a Poisson distribution is always the square root of its mean. 

Unlike non-radiometric methods of analysis, uncertainty modelling in NAA is facilitated by the existence of counting statistics, although in principle an additional source of uncertainty, because this parameter is instantly available from each measurement. If the method is in a state of statistical control, and the counting statistics are small, the major source of variability additional to analytical uncertainty can be attributed to sample inhomogeneity (Becker 1993). In other words, in Equation (2.1) ... 

The complexity of the Co emission spectrum and the low fraction of the desired 14.4 keV radiation require an efficient Mossbauer counting system that is able to discriminate photons of different energies and reject the unwanted events. Otherwise a huge nonresonant background would add to the counting statistics of the spectra and fatally increase the noise of the spectrometer. 

Mossbauer Spectral Analysis and Analog Measurements. Mossbauer spectra were obtained in the temperature range between 200 and 270 K and in two different energy windows (14.4 and 6.4 keV), which provide depth selective information about a sample [346]. To compensate for low counting statistics due to limited integration time, all available spectra were summed for the integrations on the undisturbed and brushed surface, respectively. In addition to kamacite (a-(Fe,Ni)) ( 85%) and small amounts of ferric oxide (see Fig. 8.38), all spectra exhibit features indicative for an additional mineral phase. Based on analog measurements... 

For measurements by AS, the errors of the isotope ratio will be dominated by counting statistics for each isotope. For measurements by TIMS or ICP-MS, the counting-statistic errors set a firm lower limit on the isotopic measurement errors, but more often than not contribute only a part of the total variance of the isotope-ratio measurements. For these techniques, other sources of (non-systematic) error include ... 

If the errors were greater than 10% and dominated by counting statistics, a Poisson distribution would yield slightly more accurate results. 

An accurate analysis is difficult by this technique, because the electron beam spends only a short time on each spot, and so the counting statistics will not be adequate for data of high precision to be accumulated. 

Figure 5.24(B) shows a line profile extracted from the map of Figure 5.24(A) by averaging over 30 pixels parallel to the boundary direction corresponding to an actual distance of about 20 nm. The analytical resolution was 4 nm, and the error bars (95% confidence) were calculated from the total Cu X-ray peak intensities (after background subtraction) associated with each data point in the profile (the error associated with A1 counting statistics was assumed to be negligible). It is clear that these mapping parameters are not suitable for measurement of large numbers of boundaries, since typically only one boundary can be included in the field of view.

The mapping conditions were chosen by the authors to minimize the frame time and to maximize the counting statistics by increasing the probe size and probe current, while reducing the number of pixels in the map. These authors summarized the effect of the low-magnification mapping parameters on the measured analytical resolution for their specimen/microscope combination in Table 5.1. 

Fig. 13. The total translational energy distributions for the dissociation of 03 to 0(3Pj) + 02(X%-) at 226, 230, 233, 234, 240 and 266nm. The vibrational levels of the 02(X3S ) fragment are indicated by the combs. The dotted curves represent the uncertainty in the signal intensity arising from counting statistics.

Thus for an ideal two-phase system the total calibrated intensity that is scattered into the reciprocal space is the product of the square of the contrast between the phases and the product of the volume fractions of the phases, Vi (1 — Vi) = V1V2. V1V2 is the composition parameter66 of a two-phase system which is accessible in SAXS experiments. The total intensity of the photons scattered into space is thus independent from the arrangement and the shapes of the particles in the material (i.e., the topology). Moreover, Eq. (8.54) shows that in the raw data the intensity is as well proportional to the irradiated volume. From this fact a technical procedure to adjust the intensity that falls on the detector is readily established. If, for example, we do not receive a number of counts that is sufficient for good counting statistics, we may open the slits or increase the thickness of a thin sample. 

The variations in the background, the sensitivity to moisture, the alpha activity of the chamber itself and the influence of recombination were discussed by Hultqvist. The standard deviation due to counting statistics was estimated to be about 3 % (in a few measurements 6 %). The calibration was made by counting each alpha particle by a proportional counter specially designed at the Department for this purpose. The statistical uncertainty of the calibration of the equivalent radon concentration was estimated to be 12 %. 

Figure 5. Relative standard deviation on the fitting of the deposition rate of the unattached daughters (Xun) and on the fitting of the ventilation rate (Xvent)> calculated by means of a Monte- Carlo simulation model. The lower curve is obtained with counting statistics alone. The upper curve includes one hour time fluctuations on the input parameters, with 10% rel. stand, dev. on X, un (15/h), a(.35/h), Vent(.45/h) and radon cone. (50 bq/m ) and 2% on recoil factor (.83), penetration unattached (.78) and flow rate (28 1/min).

Figure 3. The front-to-back activity ratio as measured by method 1 as function of the total wire surface area times the thickness of the screen. The numbers by the points are mesh size per inch. The error bars are calculated from counting statistics. The reason for 500 mesh having higher F/B than 635 mesh is not understood.

The counting times were chosen in such a way that at least 15 000 pulses were counted. The sorption losses were calculated from the activities of the aliquots and the activity of the aliquot taken at time zero. Taking into account the various sources of errors, mainly counting statistics, the maximum imprecision is about 3%. Therefore, calculated sorption losses of 3% and lower are omitted from the listings as being not significant. 

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