Jie's Procedure

Since the Dark Rate is the rate at which the SiPM avalanches, its value depends on 3 factors:

• The probability that an rogue electron would set off the SiPM at any one occurance

• The Rate of Scattering, or the average amount of time that is required for an electron to impact a silicon nucleus, redistributing energy randomly among the two entities.

• The number of Silicon Nuclei in the detector

The dark rate is calculated by multiplying these three factors together. It is the probability of an event, multiplied by the rate at which the probability is calculated, then multiplied by number of places that this is occuring simultaneously.

1. The actual energy of electrons follow a poisson distribution, where a large number of electrons have very little amounts of energy while a miniscule number of electrons have a very large amount of energy.

Therefore, the probability of the elections having a large amount of energy is

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Probability = e^{-E_g/kT}}

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_g Silicon = 1.12 eV}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k = 1.3806503 * 10^{-23} J K^{-1}}

1. The rate of scattering is a constant number that determines the average amount of time that is required for an electron to impact a silicon nucleus. It is dependent on two factors:
   • The average speed of an electron
   • The average distance between 2 Silicon nuclei

The electron travels at an average speed equal to the Fermi Velocity, where

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Fermi Velocity = sqrt{2E_f/M_e}}

and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Mass of Electron = 9.109 * 10^{-31} kg}

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To measure the dark rate of the SiPM,... See Earlier Experimental Procedure

The data was then compared to a predicted dark rate of the device to determine the reliability of the prediction.