The purpose of the toy Monte Carlo (TMC) event generator is to inform an accurate understanding of the effects of any signal processing techniques applied to the collected experimental data. Such an understanding is best established by the use of understood, controlled test signals which faithfully represent
the characteristics of real signals in all ways, including energy, timing, and any correlations which might exist.
C.5.1: Single photoelectron shapes
The shape of SPEs can be approximated well by the convolution of a Gaussian with an exponential decay
5. Such a shape has a closed analytical PDF described by
f(t;σ, τ, t0) = exp σ2 2τ2 − t−t0 τ 1−erf σ2−τ(t−t0) √ 2στ , (C.1)
where σis a standard parameter of the constituent Gaussian,t0 is the mean of the Gaussian, andτ is the
decay time of the exponential convolved with the Gaussian shape.
Data sheets and product manuals from Hamamatsu, the manufacturer of the PMT utilized in these experiments, provide some information on timing characteristics that might be expected of SPEs. The H11934-200 ultra-bialkali PMT used to observe the CsI[Na] crystal has approximate rise and fall times of 1.3 ns and 5.8 ns respectively when operated at -900 V bias [199, 208], slightly lower than the -950 V bias used for these measurements. Each of the photoelectrons identified via thresholding in the pretrace of waveforms from a single hour-long run were fit with a function consisting of the shape defined in Eq.(C.1) and a simple constant (baseline) offset. The fits were carried out only over a localized region of the waveform, including only 100 ns before and after the threshold crossing.
Only a small subset of collected data from the CsI[Na] measurement, collected over the course of a sin- gle hour, was used to inform representative shape parameters, The timing parameters used in the toy MC generation are fixed, rounded values informed by the fit to the parameters as described above; therefore, fluctuations in the timing characteristics of the SPE shapes are not reflected in the toy MC. Beyond fluc- tuations, it is not expected that the timing characteristics of single photoelectron pulses will appreciably change over the timescale of several days in the absence of extreme circumstances. Use of a limited range of data may also underestimate the impacts of PMT gain drift and variations in the SPE amplitude. With the simple parametrized model ultimately used to characterize the filter response in Section C.6.3, small variations in timing and SPE gain should be negligible.
C.5.2: Distribution of photoelectrons
For every event generated by the toy MC routine, the number of photoelectrons present is an unrealistic abstraction from physically realizable configurations where systematic and statistical fluctuations are largely
unavoidable. Each generated event contains a strictly specified number of photoelectrons. This number is not subjected to any statistical fluctuations: each time an event with one PE is requested, a generated waveform with a single PE is returned. Consequently, care must be taken if the toy MC routine is to be used to model very specific experimental circumstances, where energy-dependent scintillation yields, light collection effects, and statistical spreading must all be taken into account when specifying the number of PEs to include in the model waveforms.
The timing distribution of photoelectrons in the generated waveforms accounts for variation in the light production time of the CsI[Na] scintillator as well as the (comparatively small) photoelectron transit time in the PMT. Compared to the characteristic time of a single photoelectron signal from the H11934-200 PMT, the scintillation photons produced in the CsI[Na] crystal are distributed over a relatively long time. Scintillation light production has been measured by several earlier efforts, and though the fitted parameters differ, the distribution is typically modeled by an additive combination of two exponential decays, with fast and slow time constants on the order of a few hundred nanoseconds and a few microseconds. Parameters used here come from Collar et al., who report a fast decay time of τfast = 589±4 ns, a slow decay time of
τslow= 6.7±2.4 µs, and a ratio of the signal intensity between the two ofIslow/Ifast∼0.41 [77].
C.5.3: Baseline and noise “farming”
The principal motivation for a careful choice of filtering approach is the removal of subtle variations in the baseline which do not integrate to 0 through use of a mode-type baseline determination algorithm. If crucial properties of the baseline and noise were well defined and knowna priori then a targeted filter could be employed such as a high-pass filter intended to remove slowly-varying, 60-Hz line noise or a band- stop filter addressing noise of a specific frequency. In the case of the CsI[Na] experiment, however, the spectral distribution of the baseline and noise are not confined to one, or a few, narrow band(s), and the higher-frequency components overlap with the signature of SPE-like signals.
The legitimacy of any investigation of filtering that relies on artificial, generated event waveforms is predicated on accurate representation of the experimental baselines. Rather than attempting to derive a parametrized approximation of the baseline and noise structure, the data itself is called upon to provide empirical input. Since the trigger of the DAQ does not place any requirement on the presence of signal in the scattering detector (see Section 4.2.2), there are digitized events which include no photoelectrons in the scatterer signal.
These prototype events are identified by running the waveform through a level threshold algorithm, where the threshold is defined relative to a baseline determined by a mode-type approach. This technique could fail
to identify hypothetical low-amplitude signals, including them in what is considered a sample of waveforms without any signal.
When the generator routine is invoked, a “reference” data file from which baseline samples will be drawn is specified6. For each event to be generated, the generator routine samples a random number between
0 and the number of events contained in the data file supplied as a reference; the event at the randomly chosen index is checked for the presence of photoelectrons; if the randomly chosen event is found to have any non-zero number of PEs, another random number is chosen; the process repeats until an event with 0 PEs is found and this 0PE-waveform is retrieved from the reference file for use as the template upon which the simulated event is assembled. At the entrance of the event-generation routine a random number seed is determined based on the system timer, ensuring7 that subsequent event generation will not immediately
select the same baseline waveform.