Detector efficiency, calculation

It is important, first, to realize that efficiency is not a function solely of the column. Bad extracolumn parameters, such as detector cell volume or tubing diameters, can make the best column in the world look terrible. Second, efficiency measurements are very poor ways of comparing or purchasing columns unless all other parameters are constant. Many columns are bought and sold because they have a higher plate count than someone else s column. The efficiency calculations could have been made with different equations, on different compounds, on different machines, at different flow rates, all of which will have a profound effect on efficiency. The only valid use of plate counts that I have found is in column comparisons where all other variables are equal, or in following column aging over a period of days or months. 

The counting efficiency (e) of the proportional detector is calculated as the ratio of the net count rate, in s, to the activity (A), in Bq, of this standard radionuclide solution. The net count rate is the standard s gross count rate (RG) minus the detector s background count rate (RB). The reported disintegration rate (A) is the product of the radionuclide concentration, in Bq L 1, and the amount of counted sample, in L, adjusted for the radioactive decay of the radionuclide between standardization and measurement. Equation 2A.1 is the general form of this equation. 

Accurate absolute measurements rely on measured rather than calculated efficiencies. Nevertheless, an efficiency calculation is instructive because it brings forward the parameters that are important for this concept. For this reason, two cases of efficiency calculation for a photon detector are presented below. 

Efficiencies of neutron detectors are calculated by methods similar to those used for gammas. Neutrons are detected indirectly through gammas or charged particles produced by reactions of nuclei with neutrons. Thus, the neutron detector efficiency is essentially the product of the probability of a neutron interaction, with the probability to detect the products of that interaction (see Chap. 14). 

Calculate the counting rate for the case shown in the figure below. The source has the shape of a ring and emits 10 part./s isotopically. The background is zero. The detector efficiency is 80 percent, and F = 1. 

Intrinsic total detector efficiency is the probability that a gamma of a given energy which strikes the detector will be recorded. The geometry assumed for the calculation or measurement of this efficiency is shown in Fig. 12.11. 

Although it is possible in principle to theoretically calculate the element concentrations on the basis of this equation (taking into account the incident beam intensity, instrument geometry, absorption effects, detector efficiencies, etc.) (see e g., Bos and Vrielink 1998 Jenkins 1999 Hansteen et al. 2000), this is almost never done because of the inherent uncertainties in the numerous parameters. Instead various forms of standardization are utilized in an attempt to reduce the number of dependent variables. There are two general approaches (1) use a standard for each element of interest and (2) use an internal standard element. 

More recently, reliable Monte Carlo simulation of electron interactions in the source, its environment, and the detector has been used to calculate the detector efficiency curve. The efficiency curves in Fig. 7.2, calculated with the Monte Carlo n-particle code, version 4, from available beta spectral data, agree with measured efficiency values within the uncertainty of the measurements (Nichols 2006). 

Wang, Z., Kahn, B., and Valentine, J. D. 2002. Efficiency calculation and coincidence summing correction for germanium detectors by Monte Carlo simulation. IEEE T Nucl Sci 49, 1925-1931. 

The above fundamental parameter equation relates the intensity of one element to the concentration of all elements present in the sample. A set of such equations can be written, one for each element to be determined. This set of equations can only be solved in an iterative way, making the method computationally complex. Moreover, an accurate knowledge of the shape of the excitation spectrum Io E)dE, of the detector efficiency e and of the fundamental parameters //, r, w and p is required. The fundamental parameter method is of interest because it allows for semi-quantitative (5—10% deviation) analysis of completely unknown samples and is therefore of use in explorative phases of investigations. Several computer programs are available that allow one to perform the necessary calculations at various levels of sophistication. As an example, in Tab. 11.9, the relative standard deviation between certified and calculated concentration of the constituents of a series of tool steels are listed. 

The efficiency calculations were carried out by comparing the measured count rates of the detector array with the count rate observed for the ANL detector. The ratio between their... 

This count rate can be corrected for decay since irradiation ceased (exp( - Xt )), detector efficiency (E), and branching ratio (E), which is the number of y-rays emitted per disintegration of the nucleus, in order to calculate the activity of the radionuclide, Atf... 

It was shown in Section 2.3.7.1 that the absolute detection limit is a strict function of the volume and efficiency of the separation column, as well as the response factor and basehne noise of the detector. To calculate the LOD in concentration units (clod)> it was shown in Eq. 2.24 that the injected analyte mass must be taken into account. At a given analyte concentration in the sample, and in this case this relates to Clod> inj varied, by changing the injection volume... 

There have been many attempts to determine the absolute detector efficiency from various calculations. For instance, a semiempirical formula has been derived by Freeman and Jenkins (1966) and has been successfully applied by Owens et al. to HPGe detectors (Owens et al. 1991). This function was successfully fitted to the measured full-energy efficiency data (Molnar et al. 2002b), seeO Fig. 31.4. 

The calibration of a PIXE system, i.e., the determination of sensitivity factors, which assign absolute concentration data to numbers of counts in X-ray peaks, can be performed in two different ways. First, sensitivity factors can be deduced theoretically or in a semiempirical way from calculated cross sections for X-ray excitation and from X-ray absorption data for the absorbents present between the points of emission and detection, in the actual experimental setup. Second, sensitivity factors can be deduced from measurements performed on standard samples consisting of pure elements or pure chemical compounds. The detection solid angle and the energy-dependent detector efficiency should also be determined. 

When the nature and composition of the sample is not well known, it is necessary to use influence correction methods, of which there are three primary types fundamental, derived, and regression. In the fundamental approach, the intensity of fluorescence can be calculated for each element in a standard sample from variables such as the source spectrum, the fundamental eiiuations for absorption and fluorescence, matrix effects the crystal reflectivity (in a WDXRF instrument), instrument aperture, the detector efficiency, and so forth. The XRF spectrum of the standard is measured, and in an iterative process the instrument variables are refined and combined with the fundamental variables to obtain a calibration function for the analysis. Then the spectrum for an unknown sample is measured, and the iterative process is repeated using initial estimates of the concentrations of the analytes. Iteration continues until the calculated spectrum matches the unknown spectrum according to appropriate statistical criteria. This method gives good results with accuracies on the order of I %-4% but is generally considered to be less accurate than derived... 

One might imagine that, knowing all we do about the interaction processes involved, the absorption coefficients of the detector material and attenuation within the encapsulation, it would be possible to calculate the detector efficiency from first principles. Unfortunately, there are limitations in the mathematical tools at our disposal and the lack of consistency with which detectors can be manufactured militate against such calculations. At the present time, efficiency calibrations are performed on actual gamma-ray spectra. There are, however, efforts being made towards provision by the manufacturers of theoretical calibration data with each detector supplied, so that the need for calibration by the user may diminish in the future. I will discuss some of these developments in Section 7.7. 

On calculation (rather than measurement) of detector efficiency ... 

Each ion current, lots, measured with an ICP-MS instrument is biased as a result of mass discrimination, variations in detector gain and detector efficiency, and other effects. These other effects, such as background, detector dead time effects (counting devices), interferences, and matrix effects, will not be discussed here, because they vary strongly depending on the type of mass spectrometer. Discussions on these topics can be found in Chapters 2 and 3 and in the literature [38-42]. An observed isotope amount ratio, or in other words an ion current ratio, Robs.i [Eq- (6.2)], calculated for two isotopes a and b is therefore also biased. This means that every measured or observed isotope amount ratio is biased or, to put it bluntly, these observed isotope amount ratios are wrong, unless they have been corrected for all effects mentioned above. 

---