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Line 49: − + − + + + + + + + + + − A fit of this model to the data was attempted, assuming the initial peaks are Gaussians, with all parameters remaining free except the gain and the position in the spectrum corresponding to zero photo-electrons. It turned out, however, that the model could not account for the long right-hand tail of the distribution, seen in the above figure. Evidently, the naive assumption that the secondary electron emission spectrum generated by the ion is inaccurate. A crude introduction of asymmetry in the peak of secondaries was introduced by another Gaussian with its location (''r''), normalization (''β'') and width (''σ<sub>3</sub>'') as another set of free parameters.+ + + + + + + + +
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This implies a distribution in the total collected charge ''q'' from ''m'' primary photo-electrons, summed over all possible values of ''m'' weighted by its Poisson distribution:
This implies a distribution in the total collected charge ''q'' from ''m'' primary photo-electrons, summed over all possible values of ''m'' weighted by its Poisson distribution:
<math>\sum\limits_{m=0}^{\infty} \frac{e^{-\lambda} \lambda^m}{m!} f_m(q)</math>
<math>F(q) = \sum\limits_{m=0}^{\infty} \frac{e^{-\lambda} \lambda^m}{m!} f_m(q)</math>
Where ''f<sub>m</sub>'' is the ''m''-times self-convolution of the original primary and secondary peaks of the form:
Where ''f<sub>m</sub>'' is the ''m''-times self-convolution of the original primary and secondary peaks of the form:
<math>f_m(q) = [h_1(q) + h_2(q)] \
<math>f_m(q) = \left[ h_1(q) + h_2(q) \right] \otimes \left[ h_1(q) + h_2(q) \right] \otimes\,\ldots\, \equiv \left[ h_1(q) + h_2(q) \right] ^{\otimes m}</math>
A fit of this model to the data was attempted, assuming the initial peaks are Gaussians, with all parameters remaining free except the gain and the position in the spectrum corresponding to zero photo-electrons. It turned out, however, that the model could not account for the long right-hand tail of the distribution, seen in the above figure. Evidently, the naive assumption that the secondary electron emission spectrum generated by the ion is inaccurate. A crude introduction of asymmetry in the peak of secondaries was introduced by another Gaussian with its location (''r''), normalization (''β'') and width (''σ<sub>3</sub>'') as another set of free parameters.
Thus the three Gaussians generating the two initial peaks are defined as:
<math>h_1(q)=G(1,\sigma_1^2),\ h_2(q)=G(p,\sigma_2^2)+G(r,\sigma_3^2)</math>
where <math>G(\mu, \sigma^2) = \frac{1}{\sigma \sqrt{2\pi}} e^{\frac{-(x-\mu)^2}{2\sigma^2}}</math>
Since:
<math>G(\mu_1,\sigma_1^2) \otimes G(\mu_2,\sigma_2^2) = G(\mu_1 + \mu_2,\sigma_1^2 + \sigma_2^2)</math>
The, expanding the ''f<sub>m</sub>'' product using the Binomial Expansion, the distribution becomes
<math>F(q) = \sum\limits_{m=0}^{\infty} \sum\limits_{n=0}^m \sum\limits_{k=0}^{n}
\frac{e^{-\lambda} \lambda^m}{m!}
\left( \begin{matrix} m\\n \end{matrix} \right)
\left( \begin{matrix} n\\k \end{matrix} \right)
(1-\alpha-\beta)^{m-n} \alpha^{n-k} \beta^{k}
\ G\left( (m-n) + (n-k)p + kr,\ (m-n)\sigma_1^2 + (n-k)\sigma_2^2 + k\sigma_3^2 \right)
</math>
</math>