Measurement of the Longitudinal Diffusion of Ionization Electrons in the MicroBooNE Detector

The signals collected from the three wire planes undergo processing to remove the detector response²²2The measured signal can be considered a convolution of the true signal and the detector response. [9, 11] resulting in waveforms which are approximately Gaussian. Each waveform is then fitted with a Gaussian function producing a reconstructed hit. High-level reconstructed objects such as tracks, showers, and space points (three-dimensional hits) are created from collections of hits using the Pandora multi-algorithm pattern recognition software [12].

Due to the linear relationship between the squared-pulse-width in time and ionization electron drift time (equation (3.1)), it suffices to perform a linear fit of $\sigma_{t}^{2}$ versus $t$ and extract $D_{L}$ from the slope. The widths of waveforms (“pulse widths”) are sensitive to more effects than just longitudinal diffusion. Transverse diffusion, the detector response modeling, collinear delta ray production, and the angle of the reconstructed track can all significantly impact the measured time width of the pulse. To minimize the additional broadening from such effects, we place a strict set of requirements on tracks reconstructed from the MicroBooNE data.

The electron drift time in MicroBooNE ranges from 0 to 2300 $\mu$s, which we split into 25 bins. At the nominal drift velocity of 1.098 mm/$\mu$s, each bin corresponds to roughly 10 cm of drift distance. Within each of the 25 drift time bins, we employ a waveform summation technique to obtain a single representative waveform of that bin. To account for time offsets between waveforms we iteratively shift each additional waveform from -5 ticks to +5 ticks relative to the center of the summed waveform and choose the configuration which minimizes the RMS of the resultant summed waveform. An example of this process is shown in figure 8, and a sample summed waveform is shown in figure 9. The summed waveform retains a Gaussian shape, without a significant additional broadening due to the waveform summation method; see section 5.4.

Once we have a summed waveform in each bin, we fit a Gaussian to that summed waveform, taking the standard deviation as our measure of $\sigma_{t}$, and the mean as $t$. We then plot $\sigma_{t}^{2}$ vs $t$, and extract $D_{L}$ from the slope of this fit. Figure 9 shows a sample summed waveform with the Gaussian fit drawn on top. It is clear that the underlying distribution is not perfectly Gaussian, but when restricted to the region around the peak of the distribution, the Gaussian functional form is a good estimator of the width of the distribution. The statistical uncertainty on $\sigma_{t}$ is negligible due to the large number of waveforms used in each drift bin.

To validate the method described in section 3, we use simulated samples containing only a single muon. These simplified samples contain 500 events, each with exactly one muon track and no backgrounds. The muon tracks are generated precisely in-time with the beam, so there is no potential bias from $t_{0}$ mis-tagging. They populate the detector volume uniformly and have a fixed momentum of 1 GeV$/c$, with an angular coverage of $\theta_{XZ}=\pm 6^{\circ}$. Figure 10 shows the resultant plots of $\sigma_{t}^{2}$ vs. $t$ on each wire plane for simulated single muons within the angular selection values listed in section 3.1.1. For each plane, the top plot shows the linear fit and an area-normalized histogram of the number of waveforms in each bin; the bottom plot shows the fit residuals. As discussed in the previous section, each point on the plots in figure 10 represents the standard deviation of a Gaussian fit to the summed waveforms in each bin of drift time. We extract the measured $D_{L}$ value from equation (3.1) using the simulated drift velocity $v_{d}$ = 1.098 mm/$\mu$s. This simplified sample results in a measured $D_{L}$ value of 6.30 cm²/s on the collection plane. Compared to the nominal (default) simulation value of 6.40 cm²/s, the measured value is well within the estimated systematic uncertainties, discussed in section 5. The values of $\sigma_{t}^{2}(0)$ are extracted from the y-intercept of the linear fit, and their values are close to the expected value of $\sigma_{t}^{2}(0)=1.96$ $\mu\textrm{s}^{2}$. Fit errors on $D_{L}$ and $\sigma^{2}_{t}(0)$ are negligible ($<$1%).

Figure 11 shows the $\sigma_{t}^{2}$ versus drift time distribution from which we extract $D_{L}$ for MicroBooNE data. When using CRT data, the distribution of waveforms peaks near the cathode because of the CRT plane geometry; the CRT plane on the cathode side is nearly twice as large as the anode-side plane. The $D_{L}$ central value extracted from the slope is 3.74 cm²/s when using collection-plane waveforms. The statistical uncertainties and uncertainties from the fit are negligible. The y-intercept of 1.88 $\mu s^{2}$ is slightly below the expected 1.96 $\mu$s². While figure 11 shows the fit results on all three wire planes, we choose to quote the value extracted on the collection plane as our measurement. There are two primary reasons for this: 1) the induction planes are known to be more impacted by electronics noise than the collection plane, and 2) the bipolar nature of the induction plane response functions may introduce additional bias in the extracted pulse widths during deconvolution [9]. The other wire planes are used for systematic uncertainty studies as described in section 5.3. As a cross-check of this measurement, figure 12 shows area-normalized comparisons of summed waveforms between MicroBooNE data and simulated datasets with $D_{L}=$ 6.40 cm²/s (MicroBooNE nominal) and $D_{L}=$ 3.74 cm²/s (measured data value). It is clear from these comparisons that the $D_{L}=3.74$ cm²/s dataset more closely matches the data waveforms, lending weight to our measurement. Table 2 displays a summary of the results presented in figures 10 and 11.

The extracted $D_{L}$ value of 3.74 cm²/s differs significantly from the default simulation value of 6.40 cm²/s. Recall figure 16 which shows a summary of current world data on diffusion. The blue dot-dashed curve shows the parametrization of Li et al. [5] while the orange-dashed curve shows the theory prediction of Atrazhev and Timoshkin [3]. The default simulation value was extracted from the Li et al. parametrization which is known to be systematically higher than the theory curve.

A measured pulse width may have contributions to the pulse width from the effects of $D_{T}$. Adjacent electron clouds begin to overlap as they spread in the $yz$-plane under the influence of $D_{T}$, causing additional $\sigma_{t}$ smearing. Figure 13 shows an illustration of this effect. The impact of $D_{T}$ increases as a function of track $\theta_{xz}$. This motivates the strict angular requirement outlined in section 3.1.1.

To evaluate a systematic uncertainty on $D_{T}$, we generate three simulated particle gun samples using the same configuration as the sample described in section 4.1 except that we vary the simulated $D_{T}$ value in each sample. The ratio $D_{L}/D_{T}$ can be expressed as

where $\mu$(E) is the electron mobility as a function of electric field strength [5]. Following the parametrization for $\mu(E)$ given in reference [5], we find that that given our measured value of $D_{L}$ we predict $D_{T}=5.85^{+0.62}_{-0.33}$ cm²/s where the uncertainty comes from the uncertainty on the MicroBooNE E-field. We choose $D_{T}$ variation values of 4.8 cm²/s (down), 5.7 cm²/s (central value), and 7.2 cm²/s (up). These values are scaled linearly by $D_{L}^{\mathrm{Measured}}/D_{L}^{\mathrm{Simulated}}$ from the nominal simulated MicroBooNE $D_{T}$ value and uncertainties, which were designed using the Atrazhev-Timoshkin theory [3] and the available world data [4, 5].

Table 3 shows the results of running these $D_{T}$-varied samples through the $D_{L}$ analysis. The measured $D_{L}$ central values and $\sigma_{0}^{2}$ values show virtually no change when varying $D_{T}$. We attribute this to the two-dimensional nature of the MicroBooNE deconvolution—which deconvolves the signal in both time and wire space, mitigating the impact of charge spread to neighboring wires—and our stringent requirement on the value of $\theta_{xz}$. We conclude that the uncertainty on $D_{T}$ does not contribute to the systematic uncertainty on the $D_{L}$ measurement.

Equation (3.1) shows that $D_{L}$ is proportional to $v_{d}^{2}$, meaning that any uncertainty in $v_{d}$ could lead to a sizeable systematic uncertainty on $D_{L}$. MicroBooNE has measured the drift velocity across the active volume of the detector using UV laser and cosmic data [1, 10]. Across the anode plane, the drift velocity is not constant due to edge effects near the field cage. To extract $D_{L}$ from the measured slope in figure 11 (using equation (3.1)), we use $v_{d}=1.076$ mm/$\mu$s, the average value of the measured drift velocity across the anode plane.

To evaluate a systematic uncertainty on the measurement from the drift velocity, we take 1$\sigma$ variations of $v_{d}$ near the anode and recalculate $D_{L}$ using these varied $v_{d}$ values. Figure 14 shows a 2D map of the percent variation of $v_{d}$ with respect to $v_{d}$ = 1.076 mm/$\mu$s in a $yz$-slice near the anode. The drift velocity values in each bin come from the UV laser data map which was calculated using data from a dedicated calibration run in Summer 2016. Here, we ignore any bins that fall outside the waveform fiducial volume (see section 3.1.2).

Additional sources of uncertainty on $v_{d}$ include the statistical and systematic uncertainties on the drift velocity map and cosmic ray flux variations over time. Reference [10] shows that the uncertainties in the drift velocity map are dominated by statistical errors, but those errors are sub-percent level in each bin in our region of interest. The drift velocity map was calculated using laser data during the Summer of 2016, while the CRT data used in this analysis was taken between October 2017 and March 2018. Time variations of the SCE were studied in reference [1] and found to be small compared to the absolute scale of the effect. We therefore conclude that variations in SCE due to cosmic ray flux variations are already accounted for in the drift velocity map.

Because the additional sources of uncertainty discussed in the previous paragraph may be neglected, we set the uncertainty on the drift velocity by looking at the variation of the value of $v_{d}$ across our fiducial volume. The maximum $v_{d}$ variation is approximately 3% in the region near $z=400$ cm where $y<0$. However, figure 6 shows that our selected waveforms fall mostly in the region where $z>800$ cm. In this region, the drift velocity map shows that the $v_{d}$ variations are sub-percent level. Because our data mostly lay in the low-uncertainty region, We choose to conservatively apply a $\pm$2% variation to $v_{d}$. Varying the anode $v_{d}$ up and down by 2% yields variation values of 1.098 mm/$\mu s$ and 1.055 mm/$\mu s$, respectively. This difference covers any impact caused by cosmic ray flux variation and statistical uncertainties in the drift velocity map. The difference also comfortably covers any potential uncertainty from temperature variations over the data taking period, which would result in sub-percent changes in the value of $v_{d}$. Re-calculating the $D_{L}$ value shown in figure 11 using these variation values, we obtain an asymmetric drift velocity systematic uncertainty of $+3.9$%, $-4.1$%.

Equation (3.2) shows that the MicroBooNE 2D deconvolution depends on the detector response function, $R(\omega)$, as part of the deconvolution kernel. The response function has been validated on each of the three wire planes using MicroBooNE data [9], but small uncertainties on the width of the field response function⁴⁴4Recall that the response function is itself a convolution of a field response and electronics response. can have a significant impact on the width of deconvolved waveforms which in turn impacts $D_{L}$. While the final $D_{L}$ measurement uses only collection-plane waveforms, we can perform the measurement on each of the three wire planes as shown in figures 10 and 11. Since the response function on each plane was tuned independently of the others, we expect some difference in the extracted $D_{L}$ on each plane. The difference in the measured $D_{L}$ serves as a conservative estimate of the uncertainty of the wire response tuning method.

Table 2 shows that the maximum cross-plane difference in $D_{L}$ is 6.5% in the data corresponding to the difference between the V and Y planes. We therefore take 6.5% as the systematic uncertainty on the response function modeling.

We note here that, by design, applying the deconvolution to the simulation does not perfectly remove the response used during detector simulation [9]. Treating this plane-to-plane difference in the simulation analogously to the data, we are able to estimate a simulation-based response-function uncertainty of 2.5%, which covers the 1.5% difference observed on the collection plane.

The waveform summation technique described in section 3.1.2 may introduce additional broadening in the summed waveform. When aligning two waveforms, we can only shift them by integer tick values meaning that the peaks may be misaligned by as much as half a tick. We mitigate this smearing by taking the configuration which minimizes the resultant waveform RMS, but there may still be some residual broadening.

To check the impact of this effect and whether the impact is drift-dependent, we perform a study in which we sum 1000 idealized Gaussian waveforms under different conditions. We start by generating an initial Gaussian whose mean and standard deviation resemble those of waveforms from particle interactions near the anode; here, we chose “anode-like” values of $\mu=891.5$ ticks⁵⁵5MicroBooNE TPC waveforms are recorded beginning 800 ticks before the trigger time, so the position of the anode is at 800 ticks. and $\sigma=1.42$ ticks.⁶⁶6$\sigma$ here should not be confused with $\sigma_{t}$, the time width of measured signal pulses. To simulate the impact of misalignment, we also apply a random shift drawn from a uniform distribution between $-0.5$ and $+0.5$ ticks to the mean of this initial Gaussian. In the control case, we simply add this waveform to itself 1000 times using our waveform summation technique. Then, to simulate the effect of adding misaligned waveforms, we instead add 1000 waveforms with the same $\sigma$ as the initial generated Gaussian, but whose means have been shifted randomly between -0.5 and +0.5 ticks. Any difference in the extracted $\mu$ and $\sigma$ is attributed to the summation technique. We then repeat this study using “cathode-like” waveform values of $\mu=5123.4$ ticks and $\sigma=3.80$ ticks. Figure 15 and table 4 summarize the results of this study. While $\sigma$ does increase slightly in each case, the broadening is consistent at both the anode and the cathode. This may impact our extracted $\sigma_{t}^{2}(0)$ but not $D_{L}$. We repeated this study multiple times to account for different random shifts in the mean of the initial Gaussian and found no significant change in the results, including for cases where the initial Gaussian was shifted by the maximum allowed value ($\pm 0.5$ ticks). We conclude that the waveform summation technique does not introduce a sizeable systematic uncertainty to the $D_{L}$ measurement.

Other systematic uncertainties that may impact the $D_{L}$ measurement include microphysics effects that are either drift-dependent or field-dependent, particularly SCE, electron-ion recombination, and electron attenuation. For SCE, the size of the electron cloud when it arrives at the anode wire plane depends only on the amount of time that has elapsed since the electrons were ionized. We measure this time directly by using the $t_{0}$ extracted from CRT information meaning that the measurement is not biased by the presence of space charge. Thus, the measured slope of the line in figure 11 has no systematic uncertainty due to space charge. The strength of electron-ion recombination changes with the electric field, but, for MicroBooNE E-field fluctuations, this effect is small [15]. Moreover, the impact of the recombination systematic uncertainty on collected charge is much smaller than the impact of statistical Landau fluctuations in the density of ionization electron clouds. As for electron attenuation, the measured electron lifetime in MicroBooNE [16] is 18 ms. The maximum drift for a single electron is 2.3 ms meaning that charge attenuation in MicroBooNE is minimal, and this is due to the extremely high argon purity in the TPC. We conclude that both electron recombination and attenuation do not contribute to the systematic uncertainty on $D_{L}$.

Table 5 summarizes the $D_{L}$ systematic uncertainties. The two dominant systematic uncertainties come from the uncertainties on the response function modeling and the drift velocity. We have considered many other potential sources of systematic uncertainties but found them to be sub-dominant. We assume that the individual systematic uncertainties are uncorrelated and add them in quadrature to obtain the total systematic uncertainty of $+$7.6%, $-$7.7.%. This results in our final measurement from the MicroBooNE data of $D_{L}=3.74^{+0.28}_{-0.29}$ cm²/s.