1.5. METALES PESADOS EN SUELOS Y TRANSFERENCIA AL
1.5.2. ENTRADA DE METALES EN EL CURSO FLUVIAL DESDE
Now that the method has been validated on a minimally configured reduced sample, investiga- tions can be performed on a cosmic dataset. The intention is to use cosmic datasets to answer three questions:
• Are the number of drift bins and multiplication factor of the𝜎cut value which were chosen
in Section 5.3.1 still appropriate?
• Is it possible to select a sample of𝑡0-tagged tracks from a cosmic dataset which can return
the simulated𝐷𝐿value?
• What angular selection is needed to minimise the effects of𝐷𝑇?
Number of Drift Bins and𝜎 Cut Value
The plots shown in Figures 5.13 and 5.14 are reproduced using Dataset 5 as defined Table 5.1; see Figures 5.17a and 5.17b. Unlike in the case of Dataset 1, there is a significant variation in the fractional diffusion value depending on the number of bins used. There is also a small dependence on the dynamic𝜎 cut value. Because it has been shown that neither the noise or physics effects
are expected to cause significant differences in the measured 𝐷𝐿 value, it is assumed that this
dependence is due to the angular distribution of the tracks in the cosmic sample.
Because the simulated value of 𝐷𝐿 is able to be reasonably accurately measured with the
(a) Number of bins. (b)𝜎Cut Value
Figure 5.17: Fractional diffusion value as a function of the number of bins used (5.17a, using𝜎 cut
value = 1) and the𝜎 cut value (5.17b, using a number of bins of 25) for a simulated cosmic sample.
When compared with Figures 5.13 and 5.14 it is clear that the cosmic sample has more dependence on the choice of these variables, and this is assumed to be due to the angular distribution of the tracks.
Selecting𝑡0-Tagged Tracks From a Cosmic Dataset
The selection of cosmic tracks is simple. As outlined in section 5.2.1, the only requirements placed upon the tracks are a track length cut and the requirement that a track must have a reconstructed
𝑡0. As alluded to earlier, an angular selection will now be developed in order to measure𝐷𝐿using
cosmic rays.
For this analysis it is prudent to define the angular cuts in terms of the angle to the anode plane,𝜃𝑥 𝑧, and it’s orthogonal angle, 𝜃𝑦 𝑧. When using these coordinates, 𝜃𝑥 𝑧 = 𝜃𝑦 𝑧 = 90
◦ defines
a track which is going straight downwards through the detector,𝜃𝑥 𝑧 = 𝜃𝑦 𝑧 = 0
◦ is a track going
in exactly the beam direction, and𝜃𝑥 𝑧 = 𝜃𝑦 𝑧 = 180
◦is a track going exactly opposite to the beam
direction. A diagram showing these angles is provided in Figure 5.18.
The𝜃𝑥 𝑧angle, in particular, is an important angle when selecting angular cuts because of the
effect of transverse diffusion, and because the response function can have significantly different effects at different angles during deconvolution.
Figure 5.18: Plot showing the definition of𝜃𝑥 𝑧 and𝜃𝑦 𝑧. Here, the blue lines and dots represent the
collection plane, while the black and red represent the first and second induction planes.
which have been𝑡0tagged (right). The demand that tracks are𝑡0tagged reduces the statistics of
the selected dataset; however, geometric𝑡0tagging has been shown to be accurate to the order of
several ticks, as shown in Figure 5.20, and in the MicroBooNE geometric𝑡0-tagging public note
[101]. Because of this high accuracy and the large number of cosmic tracks available in any given MicroBooNE window meaning that this analysis is able to accrue statistics reasonably quickly, this is not currently deemed to not be a significant source of concern. It should be noted that the
𝑡0-tagging public note states that the purity (the fraction of tagged tracks which are truly anode-
or cathode-piercing) is approximately 97%.
Table 5.4 shows the passing rates for tracks as a function of the chosen 𝜃𝑥 𝑧/𝜃𝑦 𝑧 cut. As a
demonstration, this shows a cut which is symmetric in these variables, but this does not neces- sarily need to be the case, and will be investigated in the next section. The total passing rates appear to be very low; however, this is the passing rate on a track-by-track basis. Knowing that there are approximately 10 cosmic tracks per event increases these passing rates by a factor of
XZ θ 0 20 40 60 80 100 120 140 160 180 YZ θ 0 20 40 60 80 100 120 140 160 180 1 10 2 10 3 10 4 10 (a) Tracks>50 cm. XZ θ 0 20 40 60 80 100 120 140 160 180 YZ θ 0 20 40 60 80 100 120 140 160 180 1 10 2 10 3 10 (b)𝑡0-tagged tracks>50 cm.
Figure 5.19: Angular distribution of reconstructed tracks with a length greater than 50 cm (5.19a), and the subset of those tracks which have a reconstructed 𝑡0 (5.19b). The 𝑡0-tagging process
reduces the size of the dataset significantly more in the region of interest for this analysis (𝜃𝑥 𝑧 = 𝜃𝑦 𝑧 = 0degrees and 𝜃𝑥 𝑧 = 𝜃𝑦 𝑧 = 180degrees) than in other regions of the phase space. This is
an expected effect because the closer a track becomes to forward- or backward- going, the less likely it is to pierce the anode or cathode faces.
10. In this table, the “Selected Tracks” and “% of Total” columns are measured with respect to the total number of tracks above 50 cm, while the “𝑡0-Tagged Tracks” and “𝑡0-Tagging Rate” columns
are measured with respect to the selected tracks.
𝜃𝑥 𝑧 &𝜃𝑦 𝑧 Cut Selected Tracks % of Total 𝑡0-Tagged Tracks 𝑡0-Tagging Rate Total Pass Rate
5◦ 219 0.03% 8 3.65% 0.001%
10◦ 700 0.13% 28 4.00% 0.005%
15◦ 1568 0.29% 127 8.10% 0.023%
20◦ 3687 0.67% 456 12.37% 0.083%
Table 5.4: This table shows the total passing rate for tracks in the Monte Carlo as a function of the
𝜃𝑥 𝑧 and𝜃𝑦 𝑧 cut. This assumes that the chosen cut value is symmetric in𝜃𝑥 𝑧 and𝜃𝑦 𝑧, but this does
not necessarily have to be the case. Note that there is a requirement that every reconstructed track included in this table is at least 50 cm long.
Figure 5.20: Plots taken from the MicroBooNE𝑡0-tagging public note (reference [101]) showing
the𝑡0reconstruction accuracy for anode (left) and cathode (right) piercing tracks. As shown, the
accuracy for tracks passing through the anode is higher than for those passing the cathode, but both have a standard deviation of less than 4 ticks. For the diffusion analysis, each drift bin (for 25 bins) corresponds to approximately 184 ticks, and so it is anticipated that this resolution is more than adequate.
dataset of 30k events from Dataset 5. The result is shown in Figure 5.21. There is approximately a 1-2% bias in the central drift bins. This has been investigated and is due to the angular acceptance of each drift bin. Higher angle tracks are significantly more likely to reach the center of the TPC, and so they are more likely to be dominant in those drift bins. Although it has been shown that track angles do not affect the measurement of𝐷𝐿 for a𝐷𝑇 = 0sample up to∼30
◦, it does modify
the𝜎2
𝑡(0)values, and so a sample which uses many different track angles, such as this, is liable to
bias at the center of the TPC. For this reason, an additional demand is made that each drift bin contain more than 500 waveforms in order to be use in the linear fit. This values has been chosen by-eye. Because few low-angle tracks make it to the center of the TPC, this effectively removes the bias caused by these bins.
Dealing with the Effects of𝐷𝑇 for a Cosmic Dataset
Having now shown that the analysis is able to be performed by placing an angular cut on 𝑡0-
tagged tracks, a study has been performed to understand the bias expected in the measured𝐷𝐿
) 2 s µ ( 2 σ 1 2 3 4 5 6 7 8 /s 2 : 6.34 +/- 0.05 cm L Measured D - Fit)/Fit 2 σ ( 0.1 − 0.05 − 0 0.05 0.1 Drift Distance (cm) 0 50 100 150 200 250 #waveforms 0 1000 2000 3000
Figure 5.21: Measured 𝐷𝐿 for a cosmic sample with input 𝐷𝐿 = 6.36 cm
2/s, and
𝐷𝑇 = 0.0 cm 2/s.
Here, the number of waveforms in each drift bin is also included.
Transverse diffusion acts to modify the shape of the pulse in the𝑦 𝑧-plane. For a track which
is at 𝜃𝑥 𝑧 = 𝜃𝑦 𝑧 = 0
◦, the effect of
𝐷𝑇 is minimised because the electron cloud diffusion in the
negative𝑧-direction is compensated by the electron cloud diffusion in the positive direction. It
is not possible to achieve this using geometrically𝑡0-tagged cosmic rays because the sample is
inherently angular, and so some effective additional smearing in the longitudinal direction is present. This is shown graphically in Figure 5.22. In this graphic, the green ovals represent single electron clouds. Without𝐷𝑇, the longitudinal width of the electron cloud is measured to be the
width of a single oval in 𝑥, however after the application of 𝐷𝑇, neighbouring ovals begin to
overlap, and the measured longitudinal width is effectively wider.
In order to proceed with this measurement, a sample of tracks should be chosen which has an angular distribution which minimises the effect of𝐷𝑇, while maximising statistics. To select the
angular distribution, a study has been performed with Dataset 6, as outlined in Table 5.1. These samples use various combinations of𝐷𝑇,𝜃𝑥 𝑧, and𝜃𝑦 𝑧 to determine at which angles the effects of
MicroBooNE Preliminary
X
Y
Before D T After D TFigure 5.22: Illustration of the effects of transverse diffusion. Transverse diffusion can be thought of as a smearing in the 𝑧 direction (blue) and the𝑦 direction (orange). Because only collection
plane (vertical) wires are used, the spreading in Y has little effect because the pulses still arrive at the same time, on the same wires. However, the 𝑧 component acts to spread the cloud onto
neighbouring wires, meaning the𝑥-width of the cloud appears to gain additional width.
𝐷𝑇 become pronounced enough to significantly skew the measurement. The diffusion analysis
is performed for datasets using 𝐷𝑇 =11.3, 16.3, 21.3 cm
2/s, where the variations were chosen
because they are approximately30% away from the nominal 16.3 cm2/s, and this is expected to
be a reasonably conservative uncertainty on𝐷𝑇. For each dataset, different𝜃𝑥 𝑧 and𝜃𝑦 𝑧selections
have been made between 0 and 16 degrees. The results are presented in Figure 5.23.
For each of these figures, the plot on the left shows four bins in𝜃𝑥 𝑧and four bins in𝜃𝑦 𝑧, each
representing the percentage difference between the measured diffusion value and the true 𝐷𝐿
the average percentage bias as a function of𝜃𝑥 𝑧(green) and𝜃𝑦 𝑧(brown). It is clear from these plots
that the dominant effect on the measured value of 𝐷𝐿 is from the𝜃𝑥 𝑧 angle, with the𝜃𝑦 𝑧 angle
having little-to-no effect. For this reason, a very limited 𝜃𝑥 𝑧 angle of 0 to 4 degrees is chosen,
while the𝜃𝑦 𝑧 angle is allowed to be between 0 to 16 degrees.
These plots also reveal that, according to the model implemented in Geant4, the bias increases linearly as a function of angle, and the gradient of the 𝐷𝐿 bias as a function of 𝜃𝑥 𝑧 is increased
as the value of 𝐷𝑇 increases. This raises the interesting prospect of inferring the value of𝐷𝑇
from the measured value of𝐷𝐿 as a function of𝜃𝑥 𝑧. This is beyond the scope of this work, and
would require thought before proceeding however; one of the interesting things about diffusion in electric fields is the level of disagreement between data and theory, indicating that there may be effects which are not currently being simulated.
Due to this study, it is found that approximately a +3% bias due to 𝐷𝑇 is expected in the
nominal simulation for an angular selection of0◦≤ 𝜃𝑥 𝑧 ≤ 4 ◦and
0◦ ≤ 𝜃𝑦 𝑧 ≤ 16 ◦.
Table 5.5 summarises the final cut values chosen to perform the analysis with.
Cut Value Length 50 cm 𝜃𝑥 𝑧 0◦− 4◦ 𝜃𝑦 𝑧 0 ◦ − 16◦
Table 5.5: Cut values used to select out tracks for the diffusion analysis.