Dark Current induced trigger in MEMPHYS^{∗}

J.E Campagne - LAL - Orsay

July 18, 2006

Abstract

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1 Motivations

On 5th of July, J. Bouchez had demonstrated the need to investigate deeper than expected the dark current induced trigger for a MEMPHYS tank. As a matter of fact, a 3 MeV reactor neutrino produce around 15 p.e, so basically 15 PMTs on a Cerenkov cone are hit, while in the same time at 2 kHz dark current noise per PMT in a 400 ns time window one can expect around 160 PMT hit randomly distributed. Although, it may be not a problem (not yet proved) to recover the PMTs of the signal, the main concern is the trigger. J. Bouchez has made a preliminary numerical analysis to show that the question is relevant. A long this line, I have investigated a toy MC and analytical analysis.

2 A toy MC

The toy MC consists simply of the generation of dark current random noise on each PMT baseline, then it simulates a digital sum of all the PMT baselines, and applies a threshold on a total number of PMT hit during a sliding window. This toy MC can be adapted according to:

   NumberOfPMT  = 81,000 for MEMPHYS
   DarkCurrentRate = 6.0*kHz  more likely for 12"
   TriggerSlidingGate = 425.0*ns for MEMPHYS
   PulseWidth = 5.0*ns this is the time quantum (a pulse width equivalent)
   TimeDurationOfExp = 6000.0*micros (+TriggerSlidingGate)
                       as a total time for investigation
   CountThreshold = number of PMTs at least to be fired in a TriggerSlidingGate to
                    produce a Trigger

∗Preliminary

2.1 Dark current generation

According to the fact that dark current is a random noise, if ones make the assumption that there is no correlation between two noise hit (at least on the scale of TriggerSlidingGate), then the time interval between two hit δt follow an exponential probability with parameter τ = 1/DarkCurrentRate. So, the probability to get 1 hit in the time window [t_{0}, t_{0} + Δt] reads:

P (1; [t_{0}, t_{0} + Δt]) = ∫_{t_{0}}^{t_{0}+Δt} e^{(t−t_{0})/τ} \frac{dt}{τ}
= 1 − e^{−Δt/τ} = \frac{Δt}{τ} − \frac{1}{2} (\frac{Δt}{τ})^{2} …  (1)

and is independent of t_{0}.

If one piles up N PMTs and if Δt/τ → 0, then it is well known that the probability to get n PMTs hit in a time window of Δt tends to a Poisson distribution:

P (n|N; [t, t + Δt]) ⟶^{Δt/τ→0} Poisson(n; NΔt/τ)  (2)

One can see these properties reproduced by the toy MC on Fig. 1 for some parameters values used on purpose.

Figure 1: left panel: exponential generation of δt with 1/τ = 1 MHz dark current noise; right panel: number of PMTs fired among 50 PMTs, per Δt = 100 ns windows. One expects a Poisson distribution with a mean of NΔt/τ = 5 as it is shown.

In fact, as it is mentioned, the Poisson distribution is an asymptotic distribution. The real probability to get exactly n PMTs fired among a collection of N PMTs in a Δt time window, with the hypothesis that each PMT dark current rate has a 1/τ frequency, reads (using Eq. 1):

P (n|N; [t, t + Δt]) = C_{N}^{n}P (1; [t, t + Δt])^{n} (1 − P (1; [t, t + Δt]))^{N−n}
= Binomial(n, N; P (1, [t, t + Δt]))  (3)

2.2 Trigger simulation

One may wish to know the occurrence of ”how many times per second, n_{T h} PMTs are fired among N PMTs?”. In fact, J. Bouchez noticed that this question is not complete as one may trigger only if the number of PMTs fired in a Δt sliding windows just exceeds n_{T h}. That is to say, one should consider the stepping function of the number of PMTs fired in the sliding window Δt examined at t, noted n(t), and consider as ”trigger” the passage n(t) = n_{T h} → n(t + dt) = n_{T h} + 1 in an infinitesimal time duration dt. In fact, looking at Fig. 2, where is displayed the digital sum of N = 81, 000 PMts with 6 kHz dark current noise rate in sliding window of 425 ns (computed every dt = 5 ns), one notices that it is true that the threshold^{1} at 250 is reached by steps, but one may also specify that if there is a trigger at t_{1}, before to re-trigger the n(t > t_{1}) function may pass through a sequence n(t) = n_{T h} + 1 → n(t + dt) = n_{T h}.

Figure 2: left panel: Digital sum of the time evolution of N = 81, 000 PMTs with a dark current of 6 kHz, computed for a time sliding window of 425 ns every 5 ns; right panel: zoom around the location of a trigger threshold at 250 PMTs.

So in practice, one considers as a trigger the passage n(t) = n_{T h} → n(t+dt) = n_{T h} +1 and groups together the following n(t) values contiguous in time for which n(t) >= n_{T h}. Using this definition, one gets the results for one MEMPHYS tank (Tab.1): N = 81, 000 PMTs, a sliding window of 425 ns which corresponds to the maximum time of flight of Čerenkov light in the ϕ : 65 m × H : 65 m cylinder water volume (v = 22, 6 cm/ns).

1mean number of PMTS is 206 and the threshold is set 3 sigma above

DC per PMT (kHz) Mean of PMTs fired Threshold (at 3σ) Trigger (kHz)
1  34  52  325
2  69  93  378
3  103  133  198
4  138  172  378
5  172  211  268
6  206  249  416
DC per PMT (kHz) Mean of PMTs fired Threshold (nσ) Trigger (kHz)
1  34  40 (1)
1  34  46 (2)

Table 1: Mean number of PMTs (m)fired among N = 81, 000 PMTs on a time sliding window of Δt = 425 ns corresponding to the dark current (DC) rate (f) per PMT (this is NΔt ∗ f). Then, setting a threshold to m + 3\sqrt{m}, one reads the trigger rate (see text for the definition of a trigger).