Gain and Threshold Stability

The average instantaneous rate is typically greater at early compared to late times, and this would be the primary source of gain shifts over the course of a measuring period. The maximum rate at the beginning of the measuring period is approximately 23 kHz which is not high for a PMT-scintillator system. The average rate is approximately 4 kHz. Rate becomes an issue if two particles strike the same tile in the same measuring period. The probability for this to occur is quite low, (.04)² = 0.0016, and if necessary can be corrected for if time shifts are noted. Just after injection, we might have to deal with a slight "flash" in the forward tiles due to positrons and perhaps other sources. The positron contamination in the PSI beamlines is expected to be only a few percent and thus poses no major problem. The effect we will watch carefully is whether any time shifts occur in a PMT/base/WFD system due to a recent firing of the same detector.

The detection efficiency can change due to two causes:

• the effective discriminator is not stable; here the term discriminator extends to mean the software extraction of a real hit from the waveform records.
• the photomultiplier gain is not stable; this is a direct change of the output to a given light stimulus.

Each leads to counting errors in two ways:

1.

Some positrons will only deposit a fraction of the energy compared to a minimum ionizing particle. These are mainly positrons which range out in the second scintillator and those which strike the edges of the tiles. A shift in the effective threshold changes the detection efficiency of these positrons.

2.

Some fraction of the positrons will produce signals below the electronics threshold due to fluctuations in the number of photoelectrons. This fraction changes if the threshold level changes.

The resulting systematic error in the lifetime can be estimated by calculating the dependence of the fraction of events with a signal below threshold as a function of the threshold level.

The fraction of events stopping in the second layer can easily be evaluated using the theoretical energy distribution n(y) as given above and the fractional energy loss dE/dx for electrons from standard tables. Using this, the fraction of positrons stopping in the outer scintillator layer has been calculated for thicknesses of both layers varying from 1 mm to 20 mm and found to range up to about 0.08%. The stopping fraction depends more strongly on the thickness of the outer layer than on that of the inner layer, which suggests that the outer layer should be thinner than the inner one. For the selected thicknesses of 6.35 and 3.17 mm for the inner and outer layer, respectively, the fraction of stopped positrons is $f_{stop} = 1\times10^{-4}$. The quantity $\xi$ is the detection threshold in units of the energy deposited by a MIP. Assuming a uniform energy distribution for stopping positrons, this implies that a change in the threshold $\Delta\xi$ corresponds to a change in the fraction of stopped positrons we detect of $1\times10^{-4}\, \Delta\xi$. For small times, when drifting is expected to be most severe, the relative change in the measured lifetime is proportional to the relative change in the number of events. Therefore, the stability of the threshold $\Delta\xi$ needs to be controlled to within 1.0%, which is certainly feasible.

The fraction of events that do not stop in the second layer, but nonetheless produce a signal below threshold, depends on statistical fluctuations in the number of photoelectrons. For a large enough number of photoelectrons Gaussian statistics can be assumed, so that $\sigma_{N_{MIP}} = \sqrt{N_{MIP}}$, with N_MIP the average number of photoelectrons produced by a minimum ionizing particle. Ignoring the Landau tail of the energy deposited in the scintillator, the fraction of events below threshold is

$$\epsilon = \sqrt{\frac{N_{MIP}}{2\pi}} \int\limits_0^{\xi_... ...N_{MIP}\left(1-\frac{E}{E_{MIP}}\right)^2} \frac{dE}{E_{MIP}}$$ (8)

with $\xi$ the threshold and E_MIP the energy deposited by a MIP. In the range of 60 to 200 photoelectrons per MIP and $\xi = 0.2 E_{MIP}$, $\epsilon$ can be parametrized as $\epsilon \approx 0.36 e^{-N_{MIP}/2.77}$ and $\Delta\epsilon \approx 2.2 e^{-N_{MIP}/3.19} \Delta\xi$. For a threshold stability of $\Delta\xi = 0.01$, this means that N_MIP > 32. For N_MIP>47, $\Delta\epsilon / \Delta\xi$ is smaller than 10^-6 so that even larger threshold fluctuations do not influence the lifetime. Note that in our prototype counters, N_MIP is greater than 160.