Coincidence Timing Calibration
Introduction
During online analysis, the coincidence timing for the N-Delta experiment was calculated using the T3 and T1 trigger signals as seen from the BigBite weldment.
Figure 1: Coincidence Timing - Run 2451, Kinematic 12 - Big Bite T3-T1, full background plateau (EDTM removed) postscript | macro
These signals were also sent to both arms, and these copies could be used as well. In fact, the signal from the arms is preferred, as the common stop in the BigBite weldment is less useful for our purposes.
Figure 2: Coincidence Timing - Run 2451, Kinematic 12 - LHRS T3-T1, full background plateau (EDTM removed) postscript | macro
As can be seen in Figure 2, the coincidence timing includes a background plateau along with a coincidence peak near the center. In the plot above, which includes 150 bins across 150 ns, we can see that the background has a height of less than 6000 counts per bin, whereas the central peak has a height of 18,000 counts.
Please note that due to different travel times between the BigBite weldment and the detector hut, the coincidence peak and plateaus has shifted its timing, though the structure is similar.
Since it's the coincidence peak that is most important to us, it is worth focusing the plot on that area.
Figure 3: Coincidence Timing - Run 2451, Kinematic 12 - LHRS T3-T1, coincidence peak (EDTM removed) postscript | macro
Here we can see that the coincidence peak itself has a base width of approximately 15 ns (or a sigma of ~4.3 and a FWHM of ~9 ns). In this plot, there are 60 channels spread out over 30 ns, double the resolution of the previous plot, so the height has dropped to around half what it was above.
Comparison between triggers and scintillators
For offline analysis, it is preferrable to use the scintillator timing signals from the HRS arms themselves, specifically the signals from the S2m planes.
Normally, one would simply subtract the S2m TDCs from both arms, but due to the double-peak problems, this cannot be done. Instead, the S2m TDC signals need to be compared relative to another signal, a signal present in both arms and measured relative to each arm's own common stop.
Many signals meet this requirement. There are several signals produced in the BigBite Trigger Supervisor and sent to both arms, such as the L1A, the T1, and the T3. For this analysis, it was decided to use the T1.
While the T1 signal may not have been used as the trigger for each event, it is still present for each event, since it is created each time both the S1 and S2 bars in the right arm detect an event.
Due to differences in cable lengths between the weldment and the detector huts, the T1 signal reaches the arms at different times, but that difference is a constant and not necessarily relevant for this relative timing situation.
As such, then, the raw coincidence timing can be calculated by taking the S2m TDC values from each arm, first subtracting that arm's recorded copy of the T1 signal, and then subtracting the two results. This also has the added benefit of effectively removing the T1 signal from the calculation, leaving only the difference of the two TDCs.
To remove any time differences between the left and right PMTs on each bar, the left and right PMT TDC signals are averaged before subtracting the T1. The resulting coincidence signal can be seen below in Figure 4.
Additionally, only events in which hits were registered and TDCs recorded in S2m bars in both arms were used. This requirement eliminates any EDTM signals.
Figure 4: Coincidence Timing - Run 2451, Kinematic 12 - no offsets, full background plateau postscript | macro
The coincidence plots from the triggers and the scintillators are compared in Figure 5 below, with the blue line representing the coincidence timing from the S2m planes in the detector arms and the red line represents the coincidence timing from the triggers, which has been time-shifted by 57 ns to align with the S2m timing.
Figure 5: Coincidence Timing Comparison - Run 2451, Kinematic 12 - blue: S2m, red: DBB T3-T1 shifted 57 ns postscript | macro
In addition to the S2m peak being taller, it is worth noting that the edges of the DBB background plateau are steeper.
Again, it's worth zooming in on the peak itself, to see how it compares to the peak produced by the trigger signals.
Figure 6: Coincidence Timing Comparison - Run 2451, Kinematic 12 - blue: S2m, red: DBB T3-T1 shift postscript | macro
We see that the base width is now approximately 2 ns thinner, with a sigma of ~3.4 and a FWHM of ~8 ns, and the height of the new peak is a bit taller.
Alignment
In addition to the 1-D view, we can also compare these coincidence plots with respect to the S2m bars that are creating them.
Figure 7a: Coincidence Timing vs LHRS S2m bar - Run 2451, Kinematic 12 - no offsets, full background plateau postscript | macro
Figure 7b: Coincidence Timing vs RHRS S2m bar - Run 2451, Kinematic 12 - no offsets, full background plateau postscript | macro
Off-hand, similar plots can be produced for the trigger-based coincidence timings: LHRS and RHRS. One look at the RHRS plot will make it clear that the S2m timing is much cleaner than the DBB timing, even without offsets.
With the understanding that the colored blob represents the coincidence peak and that the 1-D peaks we've been looking at are the projections of these plots on the x-axis, it can be imagined that for best results, that colored blob should be thin, aligned, and completely vertical. Any deviation would cause our projected 1-D peak to be wider than it should be.
With this in mind, at first glance, these colored blobs may look acceptable. However, if we zoom in on the peak itself, as we did above in the 1-D case, we can see some problems.
Figure 8a: Coincidence Timing vs LHRS S2m bar - Run 2451, Kinematic 12 - no offsets, coincidence peak postscript | macro
Figure 8b: Coincidence Timing vs RHRS S2m bar - Run 2451, Kinematic 12 - no offsets, coincidence peak postscript | macro
(Again, similar plots can be created for the trigger-based timings: LHRS and RHRS.)
From the left arm, we can see a clockwise tilt as well as a misalignment in the bars, especially bar 5. And while the right arm's results aren't nearly as tilted, there is a lean toward the counter-clockwise and bar 9 is fairly misaligned.
There are three ways to correct these problems: pathlength corrections, calibration offsets, and time-walk corrections.
Pathlength Correction
The most likely cause of a tilt in the coincidence peaks relative to the LHRS S2m bars is that each bar corresponds to a different range of momenta. When the particles pass through the dipole magnet on the way to focal plane, the momentum of the particle determines how much its path will bend. The result is that particles with a lower momentum tend to hit the lower indexed bars (closer to the ceiling) and those with a higher momentum tend to hit the higher indexed bars (closer to the floor), as demonstrated in the following plot.
(Please note that while bar 15 is at the top of this plot, physically it's the lowest bar, making this, and all plots showing the S2m bars on the vertical axis, essentially upside-down.)
Lower momentum particles will take longer to reach the S2m plane, so they will arrive later, relatively speaking, than the higher momentum particles. This can clearly be seen in the Figure 8a in the previous section, where the line of coincidence peaks is tilted clockwise. The higher momentum particles, which hit the higher indexed bars, arrive first, which give them a less negative coincidence time. The lower momentum particles, which hit the lower indexed bars, arrive last, and all of this gives the LHRS coincidence timing plot a clockwise tilt.
Because the electrons' speeds are relatively unaffected by the momentum range seen in this experiment, they are all traveling at roughly the speed of light, so the coincidence timing peaks relative to the RHRS don't have much of a tilt.
To correct for this difference in momenta, we can introduce a pathlength correction, which adjusts the timing of each bar according to the momentum of the particle. Specifically, we are introducing a time offset based on the pathlength of the particle and its speed.
The distance between the target and the focal plane, which is defined to be the first VDC layer, is approximately 23.43 m and is given by the analyzer variable "X.tr.pathl", where "X" is either "L" or "R" depending on the spectrometer. Likewise, the distance between the focal plane and the S2m layer in both arms is approximately 3.14 m and given by the analyzer variable "X.s2.trpath". Together, these two variables can be added together to obtain a calculated distance between the target and the S2m layer.
As for the speed of the particle, this can be calculated from the particle's calculated momentum and its energy, which is itself calculated from the momentum and the mass. This speed, as a fraction of the speed of light, is then converted into meters per second by multiplying by the speed of light. After converting this speed into meters per nanosecond to fit with our values, the quotient of the pathlength and speed is added to the coincidence timing. The results are below.
Figure 10: Coincidence Timing - Run 2451, Kinematic 12 - pathlength corrections, coincidence peak postscript | macro
This peak has a height of over 10000 counts (though the resolution is double that of Figure 6) and a base width of approximately 6 ns (with a sigma of ~2.0 and a FWHM of ~5 ns), clearly better than the peak with no offsets. Looking at the relationship between the peaks and the S2m bars, we see something interesting.
Figure 11a: Coincidence Timing vs LHRS S2m bar - Run 2451, Kinematic 12 - pathlength corrections, coincidence peak postscript | macro
Figure 11b: Coincidence Timing vs RHRS S2m bar - Run 2451, Kinematic 12 - pathlength corrections, coincidence peak postscript | macro
(For the entire background plateau for these plots: 1-D | 2-D wrt LHRS | 2-D wrt RHRS)
The tilt appears to be gone, as expected, but the whole thing is still misaligned. To correct the misalignment, we need to apply some offsets.
Calibration Offsets
Though there are different methods for calculating calibration offsets, and many were tried, it turns out the most effective practical method was to simply look at the coincidence timing bar by bar and adjust the offsets to center the peak at a particular value. In this case, that value was 210 ns.
Average offsets from each kinematic by selecting two runs from different parts of the kinematic runtime, calculating their offsets, and averaging the resulting offsets together. The table below shows the average offsets for each kinematic.
| Bar | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 |
| Kinematic 1 | ||||||||||||||||
| LHRS | 1.59 | 0.96 | 1.00 | 0.74 | -0.34 | 1.02 | 0.25 | 0.38 | 0.47 | 1.19 | 1.00 | 2.27 | 1.88 | 3.12 | 3.32 | 4.07 |
| RHRS | 1.84 | -1.47 | -0.42 | -0.53 | -0.45 | -0.50 | 0.18 | 0.27 | 0.07 | -1.19 | 0.63 | 0.34 | 0.06 | -0.09 | 0.51 | -0.07 |
| Kinematic 2 | ||||||||||||||||
| LHRS | 1.55 | 0.71 | 0.63 | 0.37 | -0.75 | 0.61 | -0.16 | -0.02 | 0.05 | 0.77 | 0.60 | 1.81 | 1.46 | 2.78 | 2.96 | 3.68 |
| RHRS | 1.47 | -1.21 | -0.36 | -0.58 | -0.41 | -0.51 | 0.15 | 0.25 | 0.08 | -1.17 | 0.63 | 0.33 | 0.03 | -0.09 | 0.62 | -0.07 |
| Kinematic 3 | ||||||||||||||||
| LHRS | 1.84 | 0.61 | 0.59 | 0.36 | -0.76 | 0.59 | -0.15 | -0.03 | 0.04 | 0.78 | 0.62 | 1.83 | 1.46 | 2.73 | 2.94 | 3.60 |
| RHRS | 0.96 | -1.29 | -0.44 | -0.55 | -0.47 | -0.52 | 0.16 | 0.26 | 0.06 | -1.16 | 0.63 | 0.35 | 0.03 | -0.10 | 0.51 | -0.20 |
| Kinematic 5 | ||||||||||||||||
| LHRS | 2.70 | 2.07 | 2.16 | 1.92 | 0.84 | 2.21 | 1.47 | 1.34 | 1.69 | 2.41 | 2.23 | 3.49 | 3.09 | 4.33 | 4.55 | 5.12 |
| RHRS | 0.12 | -1.18 | -0.51 | -0.55 | -0.45 | -0.53 | 0.17 | 0.27 | 0.05 | -1.19 | 0.62 | 0.34 | 0.05 | -0.08 | 0.51 | -0.04 |
| Kinematic 6 | ||||||||||||||||
| LHRS | 2.14 | 1.37 | 1.39 | 1.16 | 0.13 | 1.43 | 0.70 | 0.58 | 0.91 | 1.62 | 1.44 | 2.73 | 2.31 | 3.61 | 3.72 | 4.46 |
| RHRS | 0.97 | -1.54 | -0.43 | -0.57 | -0.45 | -0.48 | 0.15 | 0.24 | 0.04 | -1.18 | 0.65 | 0.36 | 0.05 | -0.08 | 0.59 | -0.12 |
| Kinematic 7 | ||||||||||||||||
| LHRS | 2.23 | 1.40 | 1.45 | 1.19 | 0.10 | 1.46 | 0.71 | 0.81 | 0.93 | 1.67 | 1.46 | 2.73 | 2.34 | 3.62 | 3.80 | 4.48 |
| RHRS | 0.55 | -1.29 | -0.47 | -0.55 | -0.46 | -0.52 | 0.17 | 0.28 | 0.06 | -1.18 | 0.63 | 0.36 | 0.06 | -0.09 | 0.57 | -0.03 |
| Kinematic 8 | ||||||||||||||||
| LHRS | 3.50 | 2.76 | 2.75 | 2.55 | 1.43 | 2.81 | 2.09 | 1.94 | 2.25 | 3.00 | 2.80 | 4.07 | 3.70 | 4.87 | 5.07 | 5.67 |
| RHRS | -0.23 | -1.42 | -0.48 | -0.53 | -0.46 | -0.50 | 0.19 | 0.28 | 0.07 | -1.17 | 0.62 | 0.35 | 0.03 | -0.08 | 0.54 | 0.04 |
| Kinematic 9 | ||||||||||||||||
| LHRS | 3.49 | 2.33 | 2.41 | 2.26 | 1.11 | 2.50 | 1.78 | 1.48 | 1.95 | 2.68 | 2.48 | 3.76 | 3.38 | 4.60 | 4.70 | 5.41 |
| RHRS | 0.05 | -1.24 | -0.49 | -0.55 | -0.45 | -0.50 | 0.19 | 0.27 | 0.09 | -1.18 | 0.62 | 0.34 | 0.06 | -0.07 | 0.61 | -0.06 |
| Kinematic 10 | ||||||||||||||||
| LHRS | 3.44 | 2.48 | 2.45 | 2.27 | 1.16 | 2.52 | 1.78 | 1.51 | 1.98 | 2.73 | 2.51 | 3.81 | 3.39 | 4.62 | 4.82 | 5.35 |
| RHRS | 0.15 | -1.24 | -0.45 | -0.51 | -0.43 | -0.51 | 0.19 | 0.28 | 0.06 | -1.17 | 0.63 | 0.36 | 0.06 | -0.09 | 0.57 | -0.03 |
| Kinematic 11 | ||||||||||||||||
| LHRS | 2.64 | 1.85 | 1.89 | 1.65 | 0.55 | 1.91 | 1.19 | 0.92 | 1.38 | 2.12 | 1.88 | 3.18 | 2.81 | 4.03 | 4.27 | 4.83 |
| RHRS | 1.55 | -1.31 | -0.48 | -0.50 | -0.42 | -0.50 | 0.19 | 0.29 | 0.09 | -1.17 | 0.64 | 0.36 | 0.09 | -0.10 | 0.64 | -0.08 |
| Kinematic 12 | ||||||||||||||||
| LHRS | 2.64 | 1.89 | 1.89 | 1.68 | 0.58 | 1.95 | 1.20 | 1.27 | 1.40 | 2.14 | 1.95 | 3.21 | 2.85 | 4.09 | 4.23 | 4.87 |
| RHRS | 0.60 | -0.73 | -0.43 | -0.50 | -0.43 | -0.49 | 0.19 | 0.29 | 0.08 | -1.16 | 0.65 | 0.36 | 0.06 | -0.06 | 0.58 | 0.01 |
| Kinematic 13 (only one run used) | ||||||||||||||||
| LHRS | 2.64 | 1.43 | 1.36 | 1.18 | 0.07 | 1.45 | 0.69 | 0.40 | 0.88 | 1.63 | 1.40 | 2.68 | 2.31 | 3.54 | 3.69 | 4.43 |
| RHRS | 0.74 | -1.23 | -0.42 | -0.51 | -0.43 | -0.49 | 0.18 | 0.29 | 0.09 | -1.17 | 0.62 | 0.37 | 0.07 | 0.05 | 0.53 | -0.09 |
| Kinematic 14 (only one run used) | ||||||||||||||||
| LHRS | 2.64 | 2.10 | 2.07 | 1.90 | 0.79 | 2.16 | 1.43 | 1.20 | 1.60 | 2.33 | 2.14 | 3.39 | 2.99 | 4.26 | 4.43 | 5.26 |
| RHRS | -1.04 | -1.28 | -0.48 | -0.53 | -0.43 | -0.49 | 0.21 | 0.31 | 0.08 | -1.16 | 0.62 | 0.35 | 0.03 | -0.10 | 0.58 | -0.12 |
For run 2451, these offsets were then applied to the coincidence timing with the effects on the coincidence peak shown below.
Figure 12: Coincidence Timing - Run 2451, Kinematic 12 - pathlength corrections and calibration offsets, coincidence peak postscript | macro
From just this 1-D plot, we see that the height of the main peak is over 16,000 counts. Further, the base width is about 3 ns, with a sigma of ~0.7 and a FWHM of about 1.7 ns, a vast improvement over the pathlength corrections alone.
Those wavy patterns in the background is caused by the 2 ns timing structure of the beam. The fact that we can resolve that signal is an excellent sign.
And what about the alignment of the 2-D plots?
Figure 13a: Coincidence Timing vs LHRS S2m bar - Run 2451, Kinematic 12 - pathlength corrections and calibration offsets, coincidence peak postscript | macro
Figure 13b: Coincidence Timing vs RHRS S2m bar - Run 2451, Kinematic 12 - pathlength corrections and calibration offsets, coincidence peak postscript | macro
(For the entire background plateau for these plots: 1-D | 2-D wrt LHRS | 2-D wrt vs RHRS)
As can be clearly seen, the calibration offset corrected the misalignment and the pathlength correction has corrected the tilt.
But what about that third correction?
Time-walk Correction
Time-walk is the effect in which larger amplitude signals produce earlier TDC signals than smaller amplitude signals. This is due to the fact that discriminators are based on thresholds, and larger amplitude signals reach that threshold before smaller amplitude signals, even though the TDC signal should be based on the peak of the signal.
To calculate the effect due to time-walk, one examines the ADC signals with respect to the TDC signals and determines a relationship between them to adjust the timing to take it into effect.
Generally time-walk is a small effect, and in this experiment analysis, there has been discussion about simply ignoring it. To determine its worth, let us look at the relationship between the ADCs and the TDCs.
Figure 14: ADC/TDC Comparison - Run 2451, Kinematic 12 - LHRS S2m bar 7, left PMT postscript | macro
As the Figure 14 indicates, this plot is a comparison of the ADC and TDC information for the left PMT of bar 7 from the LHRS. Interestingly, we can clearly see the separate proton and pion signals, as the pion signal arrives earlier and has less energy than the proton.
But the important point here is that we see a clear tilt in the proton signal, indicating that there is some relationship between the the ADCs and the TDCs. That is, the higher the ADC channel, the higher the TDC channel, suggesting that the higher energy events are arriving at an earlier time. While this sounds similar to the pathlength correction situation, this effect is smaller and is only dealing with a single bar/PMT.
As for the electrons in the right arm, we see that this effect is almost non-existant there.
Figure 15: ADC/TDC Comparison - Run 2451, Kinematic 12 - RHRS S2m bar 7, left PMT postscript | macro
Further complicating the issue of time-walk is the absense of ADC information for the right PMTs of the left arm. Presumably the pedestal suppression was done incorrectly and all ADC information for those PMTs has been lost.
With no information for half of the left arm and no correction necessary for the right arm, calculating a time-walk correction doesn't seem possible or, given the structure seen in Figure 12, necessary.
Other Kinematics Results
To see results from the other kinematics, please click here.