Poisson Distribution/Statistics
 Practical Electron Microscopy and Database   An Online Book  

Microanalysis  EM Book http://www.globalsino.com/EM/  


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If the number of the measurements is large, the distribution can be approximately given by a Gaussian distribution. For EDS and EELS techniques in EMs, if a series of measurements are repeatedly performed on a specimen, then the distribution of counts would often follow a Poisson distribution (statistics). In other words, Poisson distribution can be employed to understand how an increase in specimen thickness affects the shape of the energy loss spectrum. In Poisson statistics, the inelastic scattering events, or collisions, are randomly independent, which states that the occurrence of one collision does not
affect the probability of the other ones. In EM measurements, the scattering of the electrons in the sample is described by the Poisson distribution, [2] Then, the fractions of electrons experiencing n order of scattering N_{n}/N_{0} as a function of relative sample thickness t/λ can be given by, Figure 3111 shows the fractions of electrons experiencing n order of scattering N_{n}/N_{0} as a function of relative sample thickness t/λ. For 1 order of scattering, it can be seen that the scattering intensity increases with thickness until about 1 mean free path where it reaches a maximum of 0.37 of the incident intensity and then decreases due to increasing number of electrons that undergo multiple scattering.
Equation 3111b can be expanded to the following equation, In principle, when only single scattering (n = 1 in Equation 3111d) exists, then the measurement is ideal and its fraction of the single scattering is proportional to the TEM sample thickness, The probability p_{n} of n plasmonloss events (or called plasmon peak or Lorentzian peak) at a specimen
thickness, t, can be given by Poisson statistics, [8] The scattering intensity associated with electrons that have suffered exactly n number of collisions is represented by I_{n}. Then, the integrated intensity
I_{n} of each plasmon peak can be given by, [8] For a limited collection semiangle β, as a result of a fortuitous property of the Lorentzian angular
distribution, one has, [11] The measured plural scattering distribution, P(E), expressed by Poisson distribution of the multiple scattering, can be given by, Then, the intensity of multiple scattering events, in Poisson distribution form, can be given by [14], Comparing with Equations 3111e and 3111i, one can obtain p_{n}(E), The p_{n}(E) is the nfold selfconvolution of the single scattering distribution p_{1}(E) and results from
all preceding scattering processes, The collection aperture accepts only a fraction F_{1} of single inelastic scattering
and a smaller fraction F_{n} of each nfold order of scattering. The various contributions to an energyloss spectrum are classfied into four
channels: [5] Fourier transformed unscattered distribution can be given by, Then, in terms of the
Fourier transformed distributions, the final distribution of electrons at all energy losses
and scattering angles, P, can be given by, [12] Note that inelastic processes are governed by Poisson statistics; however, prediction of elastic scattering by Poisson statistics is not very accurate. Equation 3111o represents the combination of the single scattering distribution and the multiple lowloss and highloss events with the elastic distribution. Equation 3111i can be rewritten by, Then, the following equation for modeling single scattering distribution can be obtained, Table 3111a. Single and multiple electron scattering.
The collection semiangle must be large enough so that the plural scattering obeys Poisson statistics. With this satisfaction, Poisson statistics can have many applications: The inevitable presence of noise in EELS spectra is mainly governed by Poisson statistics since the collection of an EELS spectrum is essentially a counting process of incoming, inelastic electrons. [10] As a result of such noise, EELS spectra made under the same conditions will differ from experiment to experiment. Table 3111b. Other EELS applications of Poisson distribution.
Background of Poisson distribution The Poisson distribution can be considered as the limiting case of a binomial distribution with np = λ, where n approaches infinity (n → ∞) and, at the same time, p approaches zero, p → 0 Additional learning: Application and validation of Poisson statistics
[1] Batson PE, Silcox J. Phys Rev B 1983;27:5224.


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