**The PM gain is found as described in the following:**

First of all, for each fixed voltage HV, we made a linear fit of the plot of the variance versus pulse height at different amplitudes of the led (the so called "Response Line"), we find thus the parameters I and S that are respectively the intersection with the Y axis and the slope, which is the Gain multiplied for (1+delta^{2}), with the right dimensions (the slope is in ADC ch., the Gain must be adimensional):G where:_{HV}=(A/e)*S/(1+delta^{2})- A = conversion factor of ADC channels
(low scale = 0.25*10
^{-12}Coulomb/chan, high scale = 0.033*10^{-12}Coulomb/chan) - e = electron charge = 1.6*10
^{-19}Coulomb - S = slope of sigma^2_peak vs peak ("Response line")
- delta = resolution of single photoelectron

The delta in the formula above has been choosed as the delta needed to obtain G(2000V) equal to the Nominal Gain from Hamamatsu (we thus normalize to Hamamatsu Gain our calibration data).

- A = conversion factor of ADC channels
(low scale = 0.25*10
**The Ideas of the PM Data Analysis:***Calibration Line*,*Working HV (HVL)*,*Error on HVL (EHVL)*

After having calculated gain at several high voltages we found a linear correlation between the gain and the HV, which is called "Calibration line" that can be written as:LogG=P2*(LogHV)+P1 *(notice that the Log is decimal, not natural)*P1 and P2 are respectively the interception with the Y axis and the slope d(LogG)/d(LogHV) (i.e. the power of the dG/G =P2dHV/HV law) of the Calibration line.

By the equation written before we could calculate the working HVL to have a certain gain G

_{0}(generally G_{0}=1x10^6):*Analitical formula to get HVL:*

HVL=(G_{0}*10^{(-P1)})^{(1/P2)}To obtain the Error on HVL (=EHVL) you cannot use the above analitical formula (it's too sensitive to decimal digits), but you can use the working formula which takes into account the HV

_{m}measured nearest to the HVL:*Working formula to get EHVL:*

Differentiating the log formula (the calibration line law) and taking a finite step, we can write:(G

_{o}-G_{m})/G_{m}=~ P2*(HVL-HV_{m})/HV_{m}The above working formula involves the HV

_{m}measured nearest to the HVL calculated and on the Gain measured at HVL (G_{0}) and at HV_{m}(G_{m}).DeltaG/G

_{m}=P_{2}*DeltaHVL/HV_{m}=>

=> EHVL=DeltaHVL=DeltaG*HV_{m}/(P_{2}*G_{m}*ln(10))