Reconstruction of the hit timing

First, we measure the ADC values of each SiPM with cosmic ray muons and prepare for a histogram of the corrected ones. Next, we prepare for another histogram of the energy deposit on the scintillator using a Monte Carlo (MC) simulation, which will be detailed in the next chapter. Then, we determine the conversion factors and the detector resolutions to match the corrected ADC values and the estimated energy deposit. Figure 3.7 depicts the histograms of the corrected values and the energy deposit estimated using the MC. The blue line in the right figure represents the fitting result. Although there exists a small discrepancy between them around the corrected ADC of 4 a.u., the line agrees with the data. Consequently, their detector resolution is 0.63±0.04 MeV, as depicted in Fig. 3.8 This resolution is consistent with the statistic estimation of the number of photons detected.

mc_0_6

Entries 51014 Mean 4.474 Std Dev 1.414

0 5 10 15 20 25 30 35 40 45 50

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10 ^mc_0_6

Entries 51014 Mean 4.474 Std Dev 1.414

mc_0_6

0 2 4 6 8 10 12 14

edep_data_var 10

102

103

Projection of conv_gaus

Data_0_6

Energy Deposit [MeV] ADCcor[a.u.]

Figure 3.7: Histograms of the ADC and the energy deposit on the scintillator. The left shows the energy deposit. The right shows the ADC measured with cosmic ray muons, and the blue line represents the fitting result.

Reconstruction of energy deposit on a layer

The hit scintillators provide the total energy deposit on a layer. To discriminate a hit on a scintillator from noise, we use an energy threshold of 0.4 MeV. Figure 3.9 depicts the reconstructed energy deposit. Here, we applied a condition that a particle hits one or adjoining two scintillators on all layers, to remove multi-hit events. We can see the MIP peaks at approximately 4 MeV, which is 10 times higher than the threshold. Therefore, the threshold is sufficiently low to discriminate hit events.

0.3 0.4 0.5 0.6 0.7 0.8 0.9 Energy Resolution [MeV]

0 5 10 15 20 Entries 25

Energy Resolution

Energy Resolution Entries 144 Mean 0.6259 Std Dev 0.03637

Energy Resolution

Figure 3.8: Histogram of the energy resolutions.

0 20 40 60 80 100 120 140

Scintillator No.

0 2 4 6 8 10 12 14 16 18 20

Energy Deposit [MeV]

0 20 40 60 80 100 120 140 160 180 200 220

edep[]:hit_seed_ch[Iteration$]+Iteration$*24 {Track_NumOfCandidates==1}

Figure 3.9: Distribution of the reconstructed energy deposit. The horizontal axis represents the scintillator channels.

wheretsc.,hitdenotes the hit time on the scintillator, ∆tprop.the delay due to photon propagation,

∆t_time-walk is the delay due to pulse-height dependence (called time walk), and ∆t_offset the offset of the measured time. The following subsections describe the methods to correct the aforementioned delay effects and reconstruct the hit timing on a scintillator.

Hit-position dependence

The photon propagation in scintillators results in a delay in the detection time on an SiPM.

First, we correct the delay. The left part of Fig. 3.10 depicts a correlation between the hit position on a scintillator and the hit timing delay of an SiPM. The correlation is fitted using the following linear function:

∆t_prop.=p₀+p₁·x_hit, (3.9)

where xhit denotes the hit position, and pi (i= 0,1) the fitting parameters. As depicted in the figure, the function describes the relation well. After correcting the dependence, we correct the other dependences.

Hit Position [cm]

TDC [ns]

Hit Position [cm]

TDC [ns]

200 400 600 800 1000

0

Figure 3.10: Hit-position dependence of the TDC. The vertical axis represents the difference between the measured time on the SiPM and the one estimated using the other layers. The horizontal axis represents the hit position.

Pulse-height dependence

When a signal crosses a threshold, the crossing time depends on the pulse height of the signal, as depicted in Fig. 3.11. This dependence is called time walk. However, the NIM EASIROC module cannot record the pulse height generated using the fast shapers. Instead of the pulse height, we use another value, which is time-over-threshold (TOT), to correct the time walk.

The TOT is the time difference between the leading edge and the trailing edges, and has strong correlation to the pulse height. Figure 3.12 depicts the correlation between the TOT and the delay of the time walk. Here, we defined the delay, ∆t_time-walk, as the time difference between the time of the leading edge and the hit time on a scintillator estimated by other layers. We corrected the delay by assuming the following function:

∆t_time-walk =p₀+ p₁

TOT−p₂, (3.10)

with three fitting parameters, p_i (i = 0,1,2). The red line on the left figure represents the fitting result, which reproduces the time walk. The right figure depicts the distribution of the difference between the time after the time walk correction and the estimated hit time

on the scintillator. Because of the time walk correction, the distribution obeys the Gaussian distribution with the mean value of zero.

threshold

T₁

time

T_det T₂

waveform1

waveform2

Figure 3.11: Outline of the time walk correction. T_det denotes the time when an SiPM detects photons, and T₁ (T₂) denotes the time when a signal of waveform1 (wavefomr2) crosses the threshold. The crossing time depends on the pulse height.

TOT [ns]

ΔT [ns]

ΔT [ns]

Figure 3.12: Result of the time walk correction. The left shows the correlation between the TOT and ∆t_time-walk. The right shows the distribution of ∆t_time-walk after the hit-position and the time walk correction.

Channel dependence

Each TDC channel has an intrinsic offset due to cable delay and other sources. As the last correction, we tuned the offset, ∆t_offset, by using cosmic ray particles. Because their velocity was almost equal to that of light, the offset was determined to match the measured hit timing to the estimated one by the other TDC values.

Reconstruction of the hit timing on scintillators

We define the reconstructed hit timing of a scintillator, t_sc.,recon, as the average time of both the SiPMs. One has the following:

t_sc.,recon = t_SiPM1,hit+t_SiPM2,hit

2 . (3.11)

The aforementioned reconstructed hit timing corresponds to t_SiPM,hit atx_hit = 60 cm, because the propagation delay is eliminated. The timing resolution of a scintillator is estimated by using the other scintillators on the other five layers. Figure 3.13 depicts the distribution of the timing resolutions of the scintillators, σ_tsc.,recon. Consequently, their resolutions are estimated to be 0.90±0.12 ns.

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

Timing Resolution [ns]

0 5 10 15 20 Entries 25

Timing Resolution

Timing Resolution

Entries 144

Mean 0.9039

Std Dev 0.1165

Timing Resolution

Figure 3.13: Distribution of the timing resolutions.

When a charged particle passes through two adjoining scintillators on a layer, the hit time is reconstructed as the weighted average of both the individual hit times. The hit time (t_recon) and its resolution (σtrecon) are given as

t_recon = t₁/σ_t²₁ +t₂/σ_t²₂

1/σ_t²₁ + 1/σ²_t2 (3.12)

and

σ_t²

recon = 1

1/σ²_t1 + 1/σ_t²₂, (3.13)

respectively. Here, we assumed the reconstructed hit time on both the scintillators, t₁ and t₂, with the resolutions of σt1 and σt2, respectively.