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 occurrence

• 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 occurring 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

$Probability=e^{-E_{g}/kT}$

where $E_{g}Silicon=1.12eV$

and $k=1.3806503*10^{-23}JK^{-1}$

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

$FermiVelocity=sqrt{2E_{f}/M_{e}}$

and $MassofElectron=9.109*10^{-31}kg$

The Average distance between 2 Silicon nuclei is known as the Mean free path of Silicon and can be determined by .......

1. The number of Silicon Nuclei in the SiPM can be obtained by calculating the total volume of the sensitive region of the SiPM multiplied by the density of Silicon

Therefore, the final prediction for the dark rate of the SiPM:

Dark Rate = $e^{-(1.12*1.6*10^{-19})/(1.381*10^{-23}*T)}$ * 0.0033 mm^3 * 200 pixels * 2.329g/1000mm^3 * 6.022*10^23molecules/28g * ($sqrt{2E_{f}/M_{e}}$) / Mean Free Path

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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.

A $X^{2}$ goodness of fit test was performed to determine how close the predicted dark rate came to the actual dark rate.

A $X^{2}$ goodness of fit test takes the

$Sum{(ActualValue-ExpectedValue)^{2}/Expected}$

If the value of the test statistic is approximately equal to the number of data points that were tested, then the equation determined above is a good predictor of the data; showing that the equation above is a good predictor of the dark rate of the SiPM used in the experiment.

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