Photon counting errors

The magnitude of the statistical errors depends on the total number of photons counted [1]. Therefore, the scanning must be carried out at an appropriately slow speed. 

Liquid scintillation counters are highly efficient for low CL intensities and consist of two photon-counting channels provided with a variable discriminator. The sample is placed between these two detectors to ensure a high optical efficiency. The discriminator is adjusted to allow photon impacts to be transmitted and small background noise pulses to be rejected. As disadvantages they suffer from saturation errors and provide nonlinear relationships between the CL intensity and the total counts. 

In absorption spectrometry, <7i is usually fairly constant, and x1 fitting has no advantages. Typical examples of data with nonconstant and known standard deviations are encountered in emission spectroscopy, particularly if photon counting techniques are employed, which are used for the analysis of very fast luminescence decays [27], In such cases, measurement errors follow a Poisson distribution instead... 

Time-correlated single-photon counting (TCSP) has proven to be a much-used method for measuring fluorescence lifetimes. It is highly sensitive in that it requires only one photon to be incident on the detector per excitation cycle, and statistical analysis of the experimental data gives lifetimes with well-defined error limits. Commercial systems are available which allow lifetimes from 50 ps to many tens of nanoseconds to be measured with relative ease and high precision. 

Obviously, the quality of intensity measurements in powder diffraction is inversely proportional to the statistical measurement errors and, therefore, it is directly proportional to the square root of the total number of registered photon counts. Assuming constant brightness of the x-ray source, the most certain way to improve the quality of the diffraction data is to use a lower scanning rate or longer counting time in continuous and step scan experiments, respectively. 

The counting error is inversely proportional to the square root of the number of photons counted for each analytical line being measured. 

The production of X-rays is a random process that can be described by a Gaussian distribution. Since the number of photons counted is nearly always large, typically thousands or hundreds of thousands, rather than a few hundred, the properties of the Gaussian distribution can be used to predict the probable error for a given count measurement. There will be a random error Sj associated with a measured intensity value I, this being equal to. As an example, if 10 counts are taken, the Is standard deviation will be [10 ] = 10 , or 0.1%. The measured parameter... 

Figure 4.4. Compaiison of two intoisity decays, on a linear (UfO and logarithmic scale irighi). The error bars te esenf Poisson noise on the photon counts. The decay functions were described in Ref. 3.

The basic act of counting a steady source of photons with any radiation detector has been shown to have associated with it a standard counting error a = /N,... 

An important limitation that is sometimes encountered is due to the particulate nature of electricity (electrons, ions) and of radiation (photons). The measurement of radiation intensities is in certain cases (e.g., X rays) performed by counting particles or photons one at a time. The number A counted in a time interval of given magnitude is subject to statistical fluctuations a count of A is subject to an estimated standard error given by... 

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