ADC to Multiplicity

The first thing to be aware of is how the translation from $ADC$ counts into $M$ is performed⁵. The ADCs for the tiles are dual range ADC, so a preliminary correction of the $ADC$s must be done:

$$ADC_i' = \left\{ \begin{array}{cl} ADC_i & \mbox{for } ADC... ...& \mbox{for } ADC_i > Gap_{i,low}\\ \end{array}\right.\quad,$$

where $Gap_{i,high}$ is the lower limit of the upper part of the ADC, and $Gap_{i,low}$ is the upper limit of the lower part of the ADC.

Next, we exclude all $ADC$s that are are lower than a certain threshold $T$:

$$T_i = P_i + a \Delta P_i\quad,$$

where $P_i$ is the ADC's pedestal, $\Delta P_i$ the pedestal width, and $a$ some constant factor. $a$ is usually set to 2.

The next step, is to subtract the pedestal from the $ADC$ values, to give us the calibrated $ADC_{cal}$:

$$ADC_{cal,i} = ADC_i' - P_i$$

As described elsewhere [2], one must make a geometrical correction for the spread in path length through a given tile, when determining the individual tile multiplicity. This correction factor $f$ is highly vertex dependent, and is given by:

$$f_i(V_z) = \sin \left(\theta_i(V_z)\right) = \frac{R}{\sqrt{(Z_i^2 - V_z^2) + R^2}}\quad,$$

where $R$ is the tile's radial distance from the beam axis, $Z_i$ the $i^{th}$ tile's position along the beam axis, and $V_z$ the primary vertex position along the beam axis.

Finally, we end up with something that is more or less proportional to the multiplicity in a given tile, by dividing by the mean deposited energy per particle $\overline{E}$ in a tile:

$$M_{i,m} = ADC_{cal,i} \frac{f_i(V_z)}{\overline{E}_i}$$ (4)

$\overline{E}$ is determined from cosmic ray data [1].