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at: ]Lawrence Livermore National Laboratory, Livermore, California 94550, USA
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# Study of dielectric breakdown in liquid xenon with the XeBrA experiment

J. Watson  University of California, Berkeley, Berkeley, California 94720, USA
Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA
<EMAIL_ADDRESS>I. Olcina  University of California, Berkeley,
Berkeley, California 94720, USA  Lawrence Berkeley National Laboratory,
Berkeley, California 94720, USA<EMAIL_ADDRESS>J. Soria  University of
California, Berkeley, Berkeley, California 94720, USA  Lawrence Berkeley
National Laboratory, Berkeley, California 94720, USA  D.N. McKinsey
University of California, Berkeley, Berkeley, California 94720, USA  Lawrence
Berkeley National Laboratory, Berkeley, California 94720, USA  S. Kravitz
Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA  E. E.
Deck  [  University of California, Berkeley, Berkeley, California 94720, USA
Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA  E. P.
Bernard  [  University of California, Berkeley, Berkeley, California 94720,
USA  Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA
L. Tvrznikova  [  University of California, Berkeley, Berkeley, California
94720, USA  Lawrence Berkeley National Laboratory, Berkeley, California 94720,
USA  Yale University, New Haven, Connecticut 06511, USA  Lawrence Livermore
National Laboratory, Livermore, California 94550, USA  W.L. Waldron  Lawrence
Berkeley National Laboratory, Berkeley, California 94720, USA  Q. Riffard  [
Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA
K.O’Sullivan  [  University of California, Berkeley, Berkeley, California
94720, USA  Lawrence Berkeley National Laboratory, Berkeley, California 94720,
USA

(August 27, 2024)

###### Abstract

Maintaining the electric fields necessary for the current generation of noble
liquid time projection chambers (TPCs), with drift lengths exceeding one
meter, requires a large negative voltage applied to their cathode. Delivering
such high voltage is associated with an elevated risk of electrostatic
discharge and electroluminescence, which would be detrimental to the
performance of the experiment. The Xenon Breakdown Apparatus (XeBrA) is a
five-liter, high voltage test chamber built to investigate the contributing
factors to electrical breakdown in noble liquids. In this work, we present the
main findings after conducting scans over stressed electrode areas, surface
finish, pressure, and high voltage ramp speed in the medium of liquid xenon.
Area scaling and surface finish were observed to be the dominant factors
affecting breakdown, whereas no significant changes were observed with varying
pressure or ramp speed. A general rise in both anode current and photon rate
was observed in the last 30 seconds leading up to a breakdown, with a marked
increase in the last couple of seconds. In addition, the position of
breakdowns was reconstructed with a system of high-speed cameras and a
moderate correlation with the Fowler-Nordheim field emission model was found.
The system of cameras also allowed us to find tentative evidence for bubble
nucleation being the originating mechanism of breakdown in the liquid. We deem
the results presented in this work to be of particular interest for the design
of future, large TPCs, and practical recommendations are provided.

††preprint: AIP/123-QED

##  I Introduction

Dielectric breakdown in time projection chambers (TPCs) is a cause of concern
because it could damage the detector and affect its overall performance.
Therefore, the characterization of dielectric breakdown in noble liquids,
especially in liquid argon (LAr) and liquid xenon (LXe), is of great
importance not only for the current generation of experiments but also for the
design of next-generation experiments such as DarkSide-20k Aalseth  _et al._
(2018), DUNE Abi  _et al._ (2018), DARWIN Aalbers  _et al._ (2016), and nEXO
Albert  _et al._ (2018). Recently, PandaX-II observed elevated photon rates
Abdukerim  _et al._ (2022), highlighting the need to understand these
phenomena.

In LXe TPCs, the drift fields are typically tens to hundreds of V/cm, and an
order of magnitude higher surrounding the high voltage (HV) feedthroughs and
in the extraction region in dual-phase TPCs. Because of this, the dielectric
strength of LXe must be well understood so that control measures, either in
the design of an experiment or during its operation, can be used to mitigate
the risk of dielectric breakdown. This is the goal of the Xenon Breakdown
Apparatus (XeBrA), located at the Lawrence Berkeley National Laboratory
(LBNL). In this work, we report on measurements of voltage breakdown in LXe
and provide an update on the work that was started in Ref. Tvrznikova  _et
al._ (2019).

The paper is organized as follows: Section I.1 outlines the main factors
affecting dielectric breakdown in LXe and Section I.2 presents the main semi-
empirical models that were explored to test these dependencies. Section I.3
introduces the Fowler-Nordheim (FN) model, used to parameterize field emission
current in our system, and Section I.4 gives a brief description of the XeBrA
experiment. Both the experimental methods and simulation analysis are
described in Section II and the body of results is presented in Section III.
The conclusions are presented in Section IV.

###  I.1 Factors affecting breakdown

A _breakdown_ is an instability which occurs when an insulating medium becomes
conducting at a sufficiently high voltage. This can happen when an electron
drifting through the medium, in addition to freeing other electrons through
primary ionization, generates one or more net electrons on average through
secondary ionization (e.g. via photoionization or positive ions impinging on
the cathodic surface). This leads to an exponential rise in current over time
that, if unstopped, culminates into a breakdown. When there is significant
electron multiplication, the space charge from the electron and ion clouds
generates an additional field. If this space charge-induced field becomes of
the same order as the bulk field, a significant rise in current occurs over a
single drift length, a phenomenon known as _streamer discharge_ Sun  _et al._
(2016). In liquid xenon, streamer discharge is believed to be the primary
breakdown mechanism due to its extremely slow ionic drift speed Hilt  _et al._
(1994).

Based on simple density scaling of gas phase data, xenon is predicted to have
a large bulk breakdown field of approximately
$1\text{\,}\mathrm{M}\mathrm{V}\mathrm{/}\mathrm{c}\mathrm{m}$ Massarczyk  _et
al._ (2017); Derenzo  _et al._ (1974) in liquid. However, LXe has been
empirically observed to break down at fields much lower than this (hundreds of
$\text{\,}\mathrm{k}\mathrm{V}\mathrm{/}\mathrm{c}\mathrm{m}$ Rebel  _et al._
(2014); Tvrznikova  _et al._ (2019)). Therefore, other mechanisms must
contribute to the positive feedback necessary to achieve dielectric breakdown.
Due to the high dielectric strength in the liquid, surface effects become the
most likely explanation. For instance, the presence of asperities on the
electrodes could enhance the local electric field and increase electron
multiplication to the point of streamer onset Phan  _et al._ (2021).

The surface area of the electrodes exposed to high fields is referred to as
the stressed electrode area (SEA). As SEA increases, the number of asperities
that could initiate a breakdown increases as well. In fact, an inversely
proportional dependence of breakdown field with stressed area has been
observed in several media Auger  _et al._ (2016); Kawashima (1974); Weber and
Endicott (1956); Goshima  _et al._ (1995a); Gerhold  _et al._ (1994). Other
dependencies have been noted as well, the main ones being:

  1. 1.

Volume: if the medium itself is a point of failure, then the breakdown voltage
would decrease in proportion to volume. This can occur in cases in which high
fields cause chemical reactions to occur or ionic bonds to dissociate. In
noble liquids, neither are factors. However, one might observe such an effect
in cases where bubbles are present throughout the stressed volume (if the
medium is very close to the boiling curve).

  2. 2.

Purity: electrons drifting in liquid attach to electronegative impurities,
such as oxygen. The reduction in charge decreases the likelihood of triggering
a breakdown.

  3. 3.

Pressure: bubble nucleation and growth are suppressed by increasing the liquid
pressure. It has been observed that increasing the liquid pressure resulted in
higher breakdown voltages in both liquid nitrogen (LN) Goshima  _et al._
(1996) and liquid helium (LHe) Gerhold  _et al._ (1996); Phan  _et al._
(2021).

  4. 4.

Ramp speed: if the risk of breakdown is represented as a differential
breakdown rate per unit time, then different ramp speeds might result in
different maximum breakdown voltages. This has been observed in borosilicate
glass and alumina Fischer and Schneider (2021), ferrofluid Bartko  _et al._
(2020), and oil-impregnated pressboardMohammed and Nedjar (2013). However, an
opposite correlation was found in thin film polymers Thomas  _et al._ (2006).

  5. 5.

Surface finish: higher sustained electric fields have been observed in more
finely polished electrodes in both LHe Gerhold  _et al._ (1994); Gerhold
(1989); Phan  _et al._ (2021) and LN2 Goshima  _et al._ (1995b). Furthermore,
acid passivation and electropolishing have been shown to be effective methods
at reducing emission rates in stainless steel wires Tomás  _et al._ (2018).

###  I.2 Hazard functions

In this section, we present a brief overview of the reliability analysis that
was performed in this work. The hazard function $h(t)$ is defined as the
differential failure rate at a given point in time $t$, conditional on
surviving up to that pointMiller (2011). Hence, the survival function $S(t)$,
defined as the probability of ramping to time $t$ without a breakdown, can be
represented as

$S(t)=\exp\left(-\int_{0}^{t}h(t^{\prime})dt^{\prime}\right)=\exp(-H(t)),$ (1)

with $H(t)$ being a dimensionless quantity referred to as the cumulative
hazard function. The cumulative hazard function can be interpreted as the
expected number of failures up to a point if trials could be restarted at
their point of failure. The hazard function fundamentally encodes how failures
occur within the system, whereas the survival function incorporates the
statistical effects of how those failures are tested.

We follow the Weibull weakest-link model to describe surface-initiated
breakdown. According to this model, the failure of a system comprised of many,
independent elements is determined by the failure of its weakest element
Weibull (1951). In such case, the failure probability density function (PDF)
as a function of electric field ($E$) can be expressed as:

$f_{W}(E;k,E_{0},E_{1})=\frac{k}{E_{0}}\left(\frac{E-E_{1}}{E_{0}}\right)^{k-1}\exp\left(-\left(\frac{E-E_{1}}{E_{0}}\right)^{k}\right),$
(2)

where $k$, $E_{0}$, $E_{1}$ are the shape, scale, and location parameters,
respectively. This is known as the three-parameter version of the _Weibull
PDF_ and the corresponding _Weibull cumulative hazard function_ follows the
power law

$H(E)=\left(\frac{E-E_{1}}{E_{0}}\right)^{k}.$ (3)

Taking $E_{1}$=0, we obtain the two-parameter version of the Weibull PDF.

As pointed out in Ref. Hill and Dissado (1983), it is possible for the
cumulative hazard function to be a function of both time and electric field,

$H(t,E)=\left(\frac{t}{t_{0}}\right)^{m}\left(\frac{E}{E_{0}}\right)^{n}=\frac{E^{m+n}}{t_{0}^{m}\dot{E}^{m}E_{0}^{n}},$
(4)

where $m$ and $n$ encode the growth rate of hazards with time and field,
respectively, and $\dot{E}$ is the ramp rate. This shows that a scan over ramp
speed is needed to determine the effect of previous breakdowns. If $m=1$, the
breakdown only depends on the field at the time of breakdown, while $m>1$ or
$m<1$ demonstrate a progressive weakening or strengthening of the dielectric
medium, respectively.

Lastly, it was mentioned in the previous section that electrode surface area
is directly proportional to the number of asperities that could initiate
breakdown. A general property of weakest link models is that the effective
hazard function of a system comprised of several elements with independent
hazard functions will be the linear combination of those hazards. Hence, when
the same electrodes are tested at different areas under the same conditions,
one arrives at the _area scaling formula_ ,

$E_{0}=C\cdot A^{-b},$ (5)

where $A$ is the SEA, $b=1/k$ is the area scaling exponent, and C is a
constant factor. This formula predicts an inverse relationship between
breakdown field and stressed area, which will be tested in this work. The
derivation of this formula can be found elsewhere Tvrznikova (2019);
Tvrznikova  _et al._ (2019).

###  I.3 Field emission

It has been suggested that breakdown does not occur in the liquid phase, but
rather that streamers in the form of small, partially ionized gas bubbles form
and then elongate along the field direction Sun  _et al._ (2016); Phan  _et
al._ (2021). These streamers disrupt the electric field surrounding them,
further enhancing the breakdown until a large, sustained current flows between
the electrodes. Field emission from the cathode provides both the initial
injection of charge as well as the heat which grows the bubbles to the
required length scale.

If this type of field emission is the primary driver of breakdowns, one can
model the hazard as being proportional to the Fowler-Nordheim (FN) field
emission current. Applying this model to stainless steel electrodes, and
following the convention from Ref. Wang and Loew (1997), the current density
($J$) is correlated to the electric field in the bulk ($E$) as

$\displaystyle J[\text{A}/\text{m}^{2}]=$ $\displaystyle\frac{1.54\times
10^{-6}}{\phi}10^{4.52\phi^{-1/2}}(\beta E)^{2}\times$ (6)
$\displaystyle\exp\left(\frac{-6.53\times 10^{9}\phi^{3/2}}{\beta E}\right),$

where $\phi$ is the LXe stainless steel work function in eV, $E$ is given in
$\text{\,}\mathrm{V}\mathrm{/}\mathrm{m}$ and $\beta$ is the local field
enhancement factor, such that $\beta E$ is the field at the tip of a field-
enhancer (e.g. an asperity). Consequently, $\log(J/E^{2})$ versus $1/E$, known
as the FN plot, is expected to show a linear relationship with a negative
slope.

###  I.4 The XeBrA experiment

XeBrA is a five-liter stainless steel chamber with adjustable, large area
electrodes that was built to investigate the factors affecting dielectric
breakdown in noble liquids. The electrodes, shown in Fig. 1, have an axially
symmetric Rogowski profile that generates highly uniform fields over a large
area, resulting in a stressed electrode area that grows with gap distance
Trinh (1980); Rogowski (1923). The anode is slightly larger ($r=56.6$ mm) than
the cathode ($r=48.4$ mm) so that a uniform electric field between the
electrons can be more easily achieved even when the two electrodes are not
perfectly aligned.Three sets of electrodes were utilized in XeBrA, two of
which were mechanically polished to four microinches, and the third underwent
citric-acid passivation.

A diagram of the XeBrA design is shown in Fig. 2. The separation between
electordes is controlled via a linear shift mechanism (LSM) with an affixed
caliper for precise control. Thus, a range of surface areas can be scanned by
simply adjusting the electrode separation. The cathode’s orientation relative
to the anode is adjusted by two set screws that are accessible through the
viewports.

A purity monitor (PM) measures the electron lifetime in-situ. Low impurity
levels are desirable in XeBrA, since the experiments XeBrA intends to inform
typically operate with very long electron lifetimes, of the order of
$1\text{\,}\mathrm{m}\mathrm{s}$ Aalbers  _et al._ (2022); Aprile  _et al._
(2022); Meng  _et al._ (2021). During a normal XeBrA run, gas circulates
through a heated zirconium getter, which removes non-noble impurities to the
part-per-billion (ppb) level. Furthermore, water and oxygen are removed from
the xenon gas with an in-line purifier of 1 part-per-million outlet purity pur
(2021).

The system is cooled through a heat exchanger (HX) coupled to a pulse tube
refrigerator. The gas inlet pressure was controlled via a proportional-
integral-differential (PID) loop operated by a programmable logic controller
(PLC) that regulates the resistive heaters affixed to the heat exchanger. Both
the high voltage ramping protocol and the resistive heaters were controlled
via a LabVIEW program Bitter  _et al._ (2006). It is worth emphasizing that
the temperature is not controlled independently of the pressure. Instead, the
stainless steel in XeBrA is typically cooled below the boiling curve in order
to suppress bubbles. The circulation speed was adjusted to tune the HX
efficiency as necessary. A diagram of the piping and instrumentation is shown
in Appendix A for extra reference.

A unique aspect of XeBrA is the presence of two viewports that frame the
electrodes, allowing visual confirmation that the liquid is in a bubble-free
state. Moreover, a pair of high-speed cameras111AOS Technologies PROMON u750
cameras are installed in each of the viewports. The cameras are held in place
with a 3D-printed mount and housing and they record at up to 2,146 frames per
second. This framerate allows each camera to capture at least one frame of the
breakdown per typical breakdown event. The two-camera system serves two main
purposes: first, to monitor the presence of bubbles in the liquid, and second,
to reconstruct the position of the breakdown in between the electrodes (see
Section III.5). More information about the XeBrA experiment can be found in
Refs. Tvrznikova  _et al._ (2019); Tvrznikova (2019).

(a)  (b)

Figure 1: _Left:_ A frontal view of one of the Rogowski electrodes. The
surface was mechanically polished and cleaned before introducing it into the
detector. _Right:_ A photograph of XeBrA’s electrodes as seen through the
front-side viewport during the filling of the chamber. Liquid xenon is visible
below the cathode. Figure 2: Diagram of the XeBrA system with its main
components annotated (see text for details). Reproduced with permission from
L. Tvrznikova, “Sub-GeV Dark Matter Searches and Electric Field Studies for
the LUX and LZ Experiments,” Ph.D. dissertation (Yale University, 2019)
Tvrznikova (2019).

##  II Methods

###  II.1 Experimental procedure

The operating loop of XeBrA consists of the cathode HV being gradually
increased until the current through the High Voltage Power Supply (HVPS)
exceeds a set threshold (typically between 1 and
$10\text{\,}\mathrm{\SIUnitSymbolMicro A}$). This causes a fault signal to be
sent by the power supply and received by the control software, which then
automatically ramps the voltage down to zero. The control software then waits
a prescribed amount of time before initiating the next ramp. During this
process, the HVPS voltage, current, system pressure, temperature, and
circulation speed are logged at a frequency of
$10\text{\,}\mathrm{H}\mathrm{z}$. The current into the anode is read out via
a charge-sensitive preamplifier and Gaussian shaping amplifier combination.

The data-taking procedure for XeBrA consists of an eight-day cycle during
which xenon is condensed, circulated, and recovered back into a high-pressure
aluminum bottle. Each cycle is labeled with a Run number and generally
contains three complete workdays of breakdown data. Two bar of gaseous xenon
are initially introduced into the chamber, while the stainless steel cools
down to xenon’s boiling point. The condensation procedure commences following
two days of circulation at 1.2–1.4 bar absolute. Approximately 12 kg of xenon
are condensed on the first day, with the fill path passing through an in-line
chemical purifier. The xenon is then circulated through a
getter222PF4-C15-R-1, manufactured by SAES Pure Gas overnight for purification
at 1.4 bar absolute. Immediately prior to the first day of breakdown data-
taking, an additional 4 kg of xenon are condensed via the circulation path,
raising the pressure to 2 bar absolute. This additional condensation rapidly
raises the condensation temperature, suppressing bubble formation near the
cathode high voltage (CHV) feedthrough.

In order to further reduce the risk of bubble formation, a “subcooling
procedure” was followed during the nights between data-taking. This procedure
consisted of lowering the set point pressure from 2.0 to 1.3–1.4 bar absolute,
while lowering the temperature on the heaters located at the gas inlet ports.
This cools the stainless steel along the boiling curve of xenon, and thermal
equilibrium is reached after several hours. The pressure is rapidly raised the
following day by reactivating the inlet heaters and increasing the speed of
the circulation pump. After the gas pressure increases to the desired set
point, a temperature inversion is established, with the bottom of the chamber
being colder than the top. The bulk of liquid xenon is then sufficiently far
away from the boiling curve to suppress bubble nucleation on the CHV
feedthrough. It was observed that this temperature inversion induced a placid
state that could be maintained for up to 8 hours, as verified by visual
inspection via the viewports.

The light produced during electrostatic discharge (ESD) is detected in several
ways. During all the runs except the first one (i.e. Runs 5–10), breakdowns
were observed with a pair of high–speed cameras, oriented approximately
perpendicular to each other, enabling position reconstruction of the breakdown
location. Cameras were not installed for Runs 1–3, corresponding to the
measurements that were presented in Ref. Tvrznikova  _et al._ (2019), and they
were not fully operative for Run 4. During Runs 5–8, the cameras were
occasionally replaced with a ONSemi 60035C-Series 6x6 mm Silicon
Photomultiplier (SiPM) sip (2021) in order to record the onset of
electroluminescence. For Runs 9–10, an additional feedthrough was installed
which enabled simultaneous camera and SiPM operation.

Data were also taken with fixed voltage in order to investigate the time
evolution of the single-photon rate. Rather than ramping until breakdown, the
cathode HV is increased to $30\text{\,}\mathrm{k}\mathrm{V}$ and maintained
there for 10 minutes. At a separation of $5\text{\,}\mathrm{m}\mathrm{m}$, it
was determined that the risk of breakdown was negligible at such voltage, and
no discharges were observed in this data-taking mode. Photons were detected
with a Photomultiplier Tube (PMT) model R9228 from Hamamatsu.

The XeBrA electrodes are negatively polarized, with the cathode receiving
negative high voltage and the anode connected to ground indirectly through a
charge-sensing electronics chain. Two surge protection circuits were placed
between the anode connection itself and a charge sensitive preamplifier. The
first was a spark gap for large surges and was placed near the feedthrough
into the inner chamber, and the second was a resistor-TVS diode circuit for
smaller surges and was placed before the amplifier. A 50 ohm coaxial cable
connected the circuits. The preamplifier was AC coupled to the grounding
circuit, allowing it to sense the fast signal from breakdowns and other
phenomena.

The amplification board consisted of a Cremat CR-150 charge sensitive
amplification board and CR-210 shaping amplifier board, with CR-111 and
CR-200-$1\text{\,}\mathrm{\SIUnitSymbolMicro s}$ amplification chips,
providing Gaussian pulse response with a full-width half maximum (FWHM) of
$2.4\text{\,}\mathrm{\SIUnitSymbolMicro s}$. The gain of the shaping amplifier
could be continuously adjusted using a potentiometer, and as such a square
wave from a function generator was used to calibrate the overall gain of the
complete electronics chain. This signal was digitized with a data acquisition
system (DAQ) from National Instruments with a sample rate of 3.5 MHz, allowing
for zero dead time acquisitions. The Polaris software Suerfu (2018) was
employed to facilitate writing DAQ data to disk.

Finally, it is worth highlighting the data structure for the analysis that is
presented. In order of least to most granular, the data collected in XeBrA can
be organized as follows: run, dataset, breakdown, and event. Runs indicate a
cycle of LXe filling and recovery, datasets indicate a particular set of data-
taking conditions (e.g. electrode gap separation and pressure), breakdowns (or
alternatively, ramps) indicate a singular high voltage ramp from
$0\text{\,}\mathrm{V}$ until discharge occurs, and events denote a division of
the digitized data, typically $280\text{\,}\mathrm{m}\mathrm{s}$ long.

###  II.2 Cathode tilt angle

Whenever the electrodes were manipulated in some way (e.g. cleaning or
exchanging them), the tilt angle and LSM caliper tare were set. The separation
was calibrated by placing a shed-free wipe between the electrodes and
squeezing them together. The wipe’s thickness was subtracted from the caliper
reading to obtain the reading that would result in a crash. A photogrammetric
approach was utilized to estimate the angle of the cathode plane relative to
the anode plane. This tilt angle has important consequences for the data
analysis described in the next sections, so a precise estimation was critical.
Photographs of the electrodes were taken from each viewport with a high-
resolution camera at known electrode separation gaps. In addition, a couple of
photographs were taken at separation distances unknown to the analyzer, which
later would be used to validate the accuracy of the method.

The tilt estimation procedure considers eight locations across the electrodes
for a given image. The LSM caliper readings which would result in the
electrodes being in contact with one another at those locations were
determined. These values were found by recording the separation distance in
pixels between the electrodes at each of the eight horizontal positions across
different photographs at various separation gaps. Then, for a given location
an orthogonal distance regression Boggs  _et al._ (1987) (ODR) was applied to
the separation gap distance as a function of the pixel separation and the
y-intercept from the fit was interpreted as the caliper reading for the
electrodes being in contact at that location.

After this process was repeated across the chosen locations, a linear
regression of the y-intercepts as a function of real-space distances was used
to find the tilt. The dimensions of a reference object in the frame were used
to scale the number of pixels into a physical distance. The radius of the
cathode is known to be 48.4 $\pm$ 0.1 mm and the number of pixels that make up
the length of the cathode can be estimated. The precision of this conversion
factor was found by applying it to the images taken at blinded separations.
The average difference between the determined separation value found using
this conversion and the known value was 0.5 mm. An additional ODR fit was then
performed to the y-intercepts as a function of real-space distances where two
sources of systematic error were considered: uncertainty in the measurement of
the cathode (0.1 mm) and uncertainty in counting pixels (0.5 mm). The slope
from the fit, along with the uncertainty on that slope, gives a tilt and its
uncertainty. These angles are shown in Table 1 for all the Runs that were
conducted in this work.

Runs | Front [∘] | Side [∘] | Total [∘]
---|---|---|---
4-5 |  0.1 $\pm$ 0.1 | -0.6 $\pm$ 0.1 | 0.6 $\pm$ 0.1
6-7 | -0.2 $\pm$ 0.1 | -0.2 $\pm$ 0.1 | 0.3 $\pm$ 0.1
8 |  0.3 $\pm$ 0.1 | -1.0 $\pm$ 0.1 | 1.0 $\pm$ 0.1
9 |  1.2 $\pm$ 0.2 | -0.4 $\pm$ 0.1 | 1.3 $\pm$ 0.1
10 | -0.1 $\pm$ 0.1 |  0.5 $\pm$ 0.1 | 0.5 $\pm$ 0.1
Table 1: Estimated cathode tilt angles for different Runs. The front and side
tilt labels refer to the tilts of the electrodes in the image plane for images
taken from the side and front viewports, respectively, and the total value is
obtained from adding those angles in quadrature.

###  II.3 Breakdown position reconstruction

In a typical data-taking run, hundreds of videos containing breakdowns were
recorded. The still frames that contain the bright flash from the streamer are
extracted for further analysis by applying an overall brightness filter. Each
of the video frames passing the filter was reviewed to ensure that the
discharge is in the frame. The frames that contain a discharge are undistorted
using OpenCV-python Bradski (2000).

The breakdown-containing frames from the individual cameras are paired using
the image timestamps. The location of the breakdown in pixel space was
recorded for each frame, corresponding to the brightest vertical strip of the
frame. The pixel coordinates of the breakdown are then transformed into the
real-space location relative to the cathode surface following a two-step
method. First, a projective transform was applied. The parameters for this
transform were found by using four fixed known points in pixel space and their
corresponding exact real space locations at the edges of the bottom electrode.
The result of this transform is the projection of two-pixel space coordinates,
coming from nearly orthogonal axes, onto the surface of the electrode. Second,
a 2nd-order polynomial transformation was applied to accurately determine the
location of the breakdown in XY space. The parameters that are used in this
second transformation were found by minimizing the residuals between positions
found after the projective transform and all corresponding known positions in
a calibration dataset. For this calibration dataset, 26 images of a target
with known geometry were taken in a test setup. In the setup, the external can
of XeBrA was removed and the target was placed at an equivalent location as
the cathode inside the can.

The result of this procedure is the real-space locations of the breakdowns on
the electrode’s surface. As an example, Fig. 3 shows an illustration of a
reconstructed breakdown in XY space along with the camera frames that were
used in the reconstruction. A solid orange line marks the location of the
brightest region in each frame, which is then used as input for the two-step
transformation. The error bars shown in that figure are calculated by
propagating the error on the location of the breakdown in the frames through
the two steps in the position reconstruction procedure.

Figure 3: Illustration of the position reconstruction analysis. The top frame
corresponds to the frontal camera and the right frame to the side camera. The
brightest regions in each frame (marked with a solid orange line) are used as
input in the transformation from pixel space to real space, resulting in the
point shown in the XY plane. The error bars reflect the systematic uncertainty
on the location of the breakdown in pixel space.

###  II.4 Electric field simulation

The finite element simulation software FEniCS Alnæs  _et al._ (2015) was used
to estimate the stressed area at different electrode separations and cathode
tilt angles. In the following, we adopt the subsequent definition of the SEA:
the area of the electrode surface with an electric field magnitude equal or
greater than 90% of the maximum electric field. The field is simulated on a
grid of separations and cathode tilt angles (see an example in Fig. 4). Note
that the SEA is a monotonically increasing function of separation distance,
growing from
$11\text{\,}\mathrm{c}\mathrm{m}\textsuperscript{\mathrm{missing}}{2}$ at
$1\text{\,}\mathrm{m}\mathrm{m}$ to
$30\text{\,}\mathrm{c}\mathrm{m}\textsuperscript{\mathrm{missing}}{2}$ at
$5\text{\,}\mathrm{m}\mathrm{m}$ (at zero tilt).

After calculating the SEA and maximum field magnitudes, the stressed area is
interpolated for the angles inferred by the photogrammetric approach described
in Section II.2. The tilt introduces a correlated uncertainty on the
calculation of the SEA at all separations. To find the uncertainty on the SEA,
a toy Monte-Carlo (MC) simulation is run, whereby several thousand cathode
tilt angles are sampled from a Gaussian distribution with mean and standard
deviations taken from Table 1. At each MC point, the SEA is interpolated from
the finite element solutions. The uncertainty on SEA is then taken as the
standard deviation of the sampled SEA values for a given configuration. The
resulting uncertainties are tabulated in Table 2.

Runs | Tilt [∘] | SEA [cm2]
---|---|---
|  | Electrode separation
|  | $1\text{\,}\mathrm{m}\mathrm{m}$ | $3\text{\,}\mathrm{m}\mathrm{m}$
4–5 | 0.6 $\pm$ 0.1 | 6.0 $\pm$ 0.7 | 16.8 $\pm$ 1.1
6–7 | 0.3 $\pm$ 0.1 | 9.6 $\pm$ 1.5 | 19.7 $\pm$ 0.8
8 | 1.0 $\pm$ 0.1 | 4.0 $\pm$ 0.3 | 12.3 $\pm$ 0.9
9 | 1.3 $\pm$ 0.1 | 3.5 $\pm$ 0.4 | 10.7 $\pm$ 1.3
10 | 0.5 $\pm$ 0.1 | 6.8 $\pm$ 1.0 | 17.8 $\pm$ 1.0
Table 2: Estimated SEA and uncertainty for each Run at two representative
separations. The effect of the tilt uncertainty is more notable in the
$1\text{\,}\mathrm{m}\mathrm{m}$ electrode separation. Figure 4: Electric
field magnitude above the cathode for a $3\text{\,}\mathrm{m}\mathrm{m}$
electrode separation and 1.3∘ cathode-anode tilt. Isocontours are indicated in
black and the color map shows the electric field distribution normalized by
its peak value. The SEA is defined as the region with an electric field
magnitude equal or greater than 90% of the peak field.

###  II.5 Purity monitor

The concentration of oxygen-equivalent impurities contained in the LXe were
measured by an in-situ purity monitor. The monitor measures the electron
lifetime as they drift between a gold-plated cathode and a brass anode. The
electrons are generated via the photoelectric effect by a xenon flash lamp,
with UV photons guided to the cathode by a fiber optic cable. Frisch grids
allow for the cathode signal to be distinguished from the anode signal. The
pulses are read out by a Cremat CR-111 charge amplifier and
CR-200-$1\text{\,}\mathrm{\SIUnitSymbolMicro s}$ shaping amplifier, generating
easily distinguishable Gaussian pulses. Data are digitized using a oscillosope
and analyzed offline. Then, the electron lifetime is found by filtering the
voltage signal with a boxcar filter and taking the ratio of the corresponding
pulse heights. This method also yields highly accurate timing information for
the pulses themselves. The ratio is converted into lifetime as

$\tau=\frac{t}{\log(A_{C}/A_{A})},$ (7)

where $t$ is drift time, $A_{C}$ is the cathode pulse area and $A_{A}$ is the
anode pulse area. The oxygen-equivalent impurity is calculated utilizing the
attachment coefficients found in Ref. Bakale  _et al._ (1976), which results
in the conversion formula

${O_{2}}=\frac{455\text{ ppb}\cdot$\text{\,}\mathrm{\SIUnitSymbolMicro
s}$}{\tau}.$ (8)

The oxygen-equivalent impurity measurements for the 2020–21 data-taking runs
are shown in Table 3. During Run 5, an overpressure incident led to xenon
mixing with residual gas in one of the recovery vessels. The purity of the
xenon after this incident was estimated using a sampling system and it was
found to be less than 100 ppb. To estimate the purity, we followed a cold trap
method similar to the one described in Ref. Leonard  _et al._ (2010). Run 6
was aborted before a purity measurement could be taken and Run 7 suffered a
data corruption issue.

The uncertainties on the purity monitor pulse areas are found by examining the
pre-cathode pulse signal RMS, while the uncertainty on the drift time is taken
to be twice the sampling period of the oscilloscope. The overall results are
tabulated in Table 3. The discrepancy in uncertainties is primarily due to
inconsistencies in the number of pulses the oscilloscope averaged over.

Run | Electron lifetime [$\text{\,}\mathrm{\SIUnitSymbolMicro s}$] | O2-equivalent impurity [ppb]
---|---|---
4 | 19.4480 $\pm$ 0.0007 | 23.3470 $\pm$ 0.0008
5-7 | - | <100
8 | 34.1 $\pm$ 0.1 | 13.34 $\pm$ 0.05
9 | 77 $\pm$ 2 | 5.9 $\pm$ 0.1
10 | 125 $\pm$ 5 | 3.6 $\pm$ 0.2
Table 3: Purity measurements taken during the 2020–21 data-taking runs. A
general trend of lower purities are observed between consecutive runs.

##  III  Results

###  III.1 Stressed electrode area scaling

At each separation and run dataset, the Weibull parameters described in
Section I.2 were extracted. The breakdowns were initially filtered to remove
confounding data. The selection criteria were: 1) breakdown voltage must be
larger than 4 kV due to occasional spurious trips on the ramp start; 2)
pressure must be within $0.05\text{\,}\mathrm{b}\mathrm{a}\mathrm{r}$ of the
set point; 3) the first ten breakdowns of any dataset are vetoed to mitigate
possible conditioning effects; 4) the breakdown must occur at least
$0.1\text{\,}\mathrm{s}$ away from a step in voltage, which ensures that we
only consider data during periods of quasi-static cathode voltage.

The top panel of Fig. 5 shows a histogram of breakdown fields for an electrode
separation of $1\text{\,}\mathrm{m}\mathrm{m}$ obtained in the first run of
this data campaign (Run 4). The breakdown field was calculated as the
breakdown voltage divided by the simulated electrode separation at the maximum
electric field (see Section II.4). Two distributions are apparent, one at
lower fields than the other, which is indicative of transient dips in
electrode performance. This feature of the data is represented by marked
corners in the cumulative hazard distribution, shown in the bottom panel and
obtained using the Kaplan-Meier estimator Kaplan and Meier (1958). Other
datasets showed a single distribution of fields, resulting in a more smooth
cumulative hazard distribution. Given this variability in the data collected,
we tested three different models for each dataset:

  1. 1.

One component, two-parameter Weibull PDF

  2. 2.

One component, three-parameter Weibull PDF

  3. 3.

Two components, three-parameter Weibull PDF

Model 3 is constructed as a normalized linear combination of two Weibull PDFs.
Model 1 is treated as the null hypothesis initially, and against it the
likelihood ratio of Models 2 is tested:

$\displaystyle\ln\left(\frac{\mathcal{L}_{2}}{\mathcal{L}_{1}}\right)=$ (9)
$\displaystyle\sum_{i=1}^{N}\left(\ln
f_{W}(E_{i};k^{(2)},E_{0}^{(2)},E_{1}^{(2)})-\ln
f_{W}(E_{i};k^{(1)},E_{0}^{(1)})\right).$

The subscript $i$ runs over data points while the superscript indicates the
corresponding Weibull parameters of each model. Model 1 is rejected if the
$p$-value of this likelihood ratio scores below 5%. In that case, the process
is repeated for Model 2 and Model 3, with Model 2 as the null hypothesis. With
this method we test if progressively adding more parameters to the model
improves the fit to data significantly. If Model 3 is selected, the best-fit
parameters from the larger component are chosen for further analysis. The
reason for this choice is that we believe the best-fit parameters from the
higher component are more representative of the breakdown mechanism under
study. The lower component in those fits are most likely due to temporary
instabilities of the system caused by the presence of debris or bubbles,
transient changes in the thermodynamics, or surface pitting. These low
breakdown field events were clustered in specific periods of time, suggesting
that they are not a symptom of some underlying condition.

Large individual uncertainties on the scale ($E_{0}$) and location ($E_{1}$)
parameters are present when Model 2 or 3 were selected. Also, a high degree of
anti-correlation between those two parameters was observed. This is expected,
since statistical quantities such as the mean and variance of the Weibull PDF
depend on all of the model parameters. Combinations of those two parameters
were used to provide more statistically robust and interpretable metrics. In
particular, the shifted scale parameter $E_{2}\equiv E_{0}+E_{1}$ was used
since it has approximately the same interpretation as the scale parameter of a
two-parameter Weibull PDF (i.e. the 63rd percentile of the distribution). One
consequence of the large anti-correlation observed between the $E_{0}$ and
$E_{1}$ parameters is that the off-diagonal coefficients of the covariance
matrix become sufficiently large to play a relevant role in the propagation of
uncertainty, resulting in a lower uncertainty overall for the shifted scale
parameter $E_{2}$. The best-fit parameters for all the datasets taken in this
campaign are shown in the appendix (see Table 6).

An alternative approach to fit the data would be to use a linear combination
of cumulative hazard functions, $H(E)$, but that was observed to return
generally worse fits for the datasets where deviations from linearity were
seen. This can be interpreted as indicating that a transient hazard was
observed, rather than simultaneous hazards (e.g. the flat region shown in the
bottom panel of Fig. 5 cannot be modelled with a linear combination of
cumulative hazard functions.)

Figure 5: An example of a dataset that strongly favors a two component, three-
parameter Weibull PDF (i.e. Model 3, see text). This is for Run 4 and
electrode separation 1 mm. _Top:_ Histogram of breakdown fields (blue) and
two-parameter Weibull PDF overlaid on top. _Bottom:_ The cumulative hazard
distribution (black) and the model that provided the best fit to data (orange
dash line). $E_{2}$ is equal to the shifted scale parameter $E_{0}+E_{1}$,
corresponding to the 63rd quantile of the distribution, and is plotted here
for reference (black dash). The cumulative hazard can be interpreted as the
expected number of breakdowns that need to occur before reaching a given
breakdown field, if ramps were restarted at the point of failure.

The results of all the fits to data are shown in Fig. 6. The “M. Polished
(2020)” fit was performed on datasets from Runs 4, 6, 7, and 9, in which
several electrode separations were tested (and therefore SEA values), using a
pair of mechanically polished electrodes. Runs 5 and 8 were excluded from the
fit: the former because of an overpressure event during condensing and the
latter due to the use of a different set of electrodes with which significant
different results were observed. Run 9 was performed with the same set of
electrodes as Runs 4–7, but they were re-polished because it was observed that
the previous experimental runs had resulted in pitting of the electrodes (see
Section III.6). Additionally, a fit was performed on the datasets
corresponding to the passivated electrodes (Run 10), which is labelled as
“Passivated”. These electrodes, which were passivated with citric acid, were
observed to outperform the mechanically polished electrodes, especially at
smaller SEA. However, we note that this fit is based on three data points and
a finer sampling of SEA, as well as an extension to higher values, is needed
to verify this trend. Overall, a downwards trend in breakdown field with
stressed area was observed in all cases.

The best-fit parameters to the power law in Eq. 5, as applied to $E_{2}$,
along with their Pearson correlation coefficient, are tabulated in Table 4.
The error bars correspond to the quadratic sum of the statistical uncertainty,
taken from the covariance matrix of the Weibull fit parameters, and a
systematic uncertainty propagated from the systematic uncertainty in the
measurement of the electrode separation ($0.03\text{\,}\mathrm{m}\mathrm{m}$).
The errors in SEA are obtained via the FEniCS simulation after accounting for
the cathode tilt angle at each run (see Section II.4). The summed squares of
residuals along both axes were minimized to obtain these fits.

One of the current generation liquid xenon TPCs is the LUX-ZEPLIN (LZ)
experiment. The LZ cathode ring has a simulated stressed area of approximately
940 cm2. Evaluating the model from this work for the mechanically polished
electrodes gives a predicted shifted scaled parameter ($E_{2}$) of 28.2 $\pm$
0.5 kV/cm. This is a revision downwards from the prediction in Ref. Tvrznikova
_et al._ (2019), shown in blue in the top figure of Fig. 6. Using the model
from XeBrA’s 2018 analysis, the predicted breakdown field at the LZ cathode
ring SEA is $70.4\pm 3$ kV/cm. We believe the discrepancy results mainly from
the following changes: the uncertainty on the cathode tilt angle was taken
into account (instead of assuming a 0∘ tilt of the cathode), each run
consisted of several days of operation in which the xenon was continuously
being circulated (as opposed to one very long day of data-taking), and a new
method to mitigate bubble formation in the bulk was introduced. To provide
some reference, the maximum design voltage in LZ was set at 50 kV/cm Mount
_et al._ (2017), which is intermediate between these two extrapolated values.
In addition, the predicted breakdown voltage for a 40-tonne active volume TPC
(e.g. DARWIN Aalbers  _et al._ (2016)), assuming that SEA scales with mass to
the power of $2/3$, would be $20.7\pm 0.4$ kV/cm. This is illustrated in the
bottom panel of Fig. 6, where the estimated stressed area of the cathode ring
for LZ and two hypothetical Generation-3 (G3) experiments are shown.
Consequently, careful engineering of the electrostatics of any future, large
TPC will be critical in achieving the desired fields. This analysis indicates
that future LXe experiments of the scale of DARWIN should design around a
maximum electric field of
$20\text{\,}\mathrm{k}\mathrm{V}\mathrm{/}\mathrm{c}\mathrm{m}$ on their
cathode rings in order to have some safety margin. However, we note that
improvements on cathode design could raise this threshold.

Figure 6:  The dependence of breakdown field on stressed area. The Weibull scale plus location parameters are plotted ($E_{2}\equiv E_{0}+E_{1}$). This statistic was chosen due to the high level of correlation between the scale ($E_{0}$) and location ($E_{1}$) parameters. Top: The current stressed area scaling using mechanically polished (black) and passivated electrodes (red) are compared to data from the 2018 analysis (blue), in which mechanically polished electrodes were used. Although the data points are taken at identical separation distances, there are more stressed areas displayed in the 2020 measurements due to the different estimated cathode tilt angles between some runs. The blue data points were reproduced with permission from JINST 14 P12018 (2019) Tvrznikova  _et al._ (2019). Copyright IOP Publishing, Ltd. Bottom: The 2020 mechanically polished data and polynomial fit extrapolated to larger SEA values. In addition, the estimated SEAs for LZ (7 tonnes active LXe), and two possible next-generation TPCs (with 40 and 80 tonnes active LXe, respectively) are shown.  Model | C [kV/cm] | b | $\rho_{Cb}$
---|---|---|---
M. Polished (2020) | 169.5 $\pm$ 3 | 0.262 $\pm$ 0.006 | -0.995
Passivated (2020) | 530 $\pm$ 28 | 0.667 $\pm$ 0.02 | -0.920
M. Polished (2018) | 171.5 $\pm$ 8 | 0.13$\pm$ 0.02 | -0.995
Table 4: XeBrA power law fit parameters to Eq. 5. $C$ is a multiplicative
constant, and $-b$ is the coefficient of the SEA in the power-law scaling.

Lastly, and while not central to this analysis, the theory that field emission
from local asperities initiates dielectric breakdown remains a possible
interpretation of the data. Without a high precision direct measurement of DC
current, the area of the field emitter can not be unambiguously determined.
Following the convention in Ref. Phan  _et al._ (2021), we make the assumption
that the cumulative hazard is proportional to the FN field emission current.
In such case, the slope of the FN plot ($\log(H(E)/E^{2})$ vs $1/E$) combined
with the work function of stainless steel provides a measurement of $\beta$,
the local field enhancement factor.

An example of such plot for Run 7 (using polished electrodes) is provided in
Fig. 7. Deviations from linearity are observed at lower fields and are not
included in the final fit. A linear regression performed on the quasi-linear
regions of the highest fields, yields $\beta$ values ranging from $100$ to
$2000$ for the different electrode separations that were scanned. Such
disparity among the estimated $\beta$ values implies the possibility of
emitter locations changing over the course of the run. Similar results were
observed for different runs. Albeit not impossible, this is an unlikely
result. Consequently, the FN model was not utilized in the final analysis, but
field emission from the cathode remains a possible explanation for a starting
mechanism of dielectric breakdowns in the liquid.

Figure 7: Fowler Nordheim plot for Run 7. $H(E)$ represents the cumulative
hazard due to the electric field. All datasets shown here have a pressure
setpoint of 1 bar absolute and a ramp speed of
$100\text{\,}\mathrm{V}\mathrm{/}\mathrm{s}$. An increase in hazard,
identified as a pseudo-current ($J$), occurs as the separation distance
increases. Black dashed lines represent a power-law fit to the linear regions
at high field.

###  III.2 Pressure

Liquid pressure was scanned in addition to separation distance. Pressure
values ranging from 1.5 to 2.2 bar absolute were scanned across all runs at a
fixed ramp rate of $100\text{\,}\mathrm{V}\mathrm{/}\mathrm{s}$ (see the
complete list of configurations in Table 6). The separation distance was fixed
at $2\text{\,}\mathrm{m}\mathrm{m}$ and the pressure was adjusted in steps. No
unambiguous relationship between the system pressure and breakdown voltage was
observed for any of the runs. An approximately $10$% variation in breakdown
field between datasets at a different pressure was observed, however this
result was insensitive to the direction of change. During one of the runs (Run
8) the pressure was increased monotonically and the breakdown field was
observed to be more consistent, yet still largely insensitive to pressure
changes.

We suspect that the overall thermodynamics of the system have a larger impact
on the breakdown field than anticipated. Pressure has a direct dependence on
the temperature and circulation flow rate in the system, making it difficult
to control independently. Changes to the operational procedure of XeBrA might
help to control for some of these confounding variables better in the future.

###  III.3 Ramp speed

If breakdown risk were modeled as a rate per unit time, it would be expected
that a faster ramp speed would lead to larger breakdown fields. As Eq. 4
shows, breakdowns cannot generally be described as a simple threshold. By
contrast, there is some stochastic risk of failure per unit time, and
increasing the high voltage faster entails spending less time at any
particular field. We caution that the experimental framework here does not
describe the operating conditions of typical TPCs, where the voltage is ramped
to a specified voltage and held there for potentially years. Instead, what Eq.
4 does is to disambiguate the variance of the breakdown values (i.e. the
Weibull shape parameter) from the effect of the “history” of each ramp. An
example of this would be the accumulation of damage to the electrodes or the
medium from previous discharges.

In XeBrA, a larger breakdown field was indeed observed with increasing ramp
speed with passivated electrodes, but inconclusive results were found with the
non-passivated, polished electrodes. In the range of 100 to 150 V/s, both sets
of electrodes show an increase in breakdown field of comparable magnitude. At
$200\text{\,}\mathrm{V}\mathrm{/}\mathrm{s}$ the positive trend continues for
the passivated electrodes but reverses for the polished electrodes.

Hence, no definitive model of breakdown fields as a function of ramp speed was
found. The observed variation between datasets with different ramp speeds is
approximately 10%. Future runs of XeBrA will explore more values of the ramp
speed scan than the ones presented in this work.

###  III.4 Precursors

It is of particular interest to determine whether an imminent breakdown can be
sensed, such that possible interventions may be taken to avoid damage to the
detector. If breakdowns are initiated by field emission near local asperities,
one might expect an increase in current into the anode as the cathode voltage
increases, as both the FN current and any multiplication effects increase in
magnitude. In XeBrA, precursor discharges were typically observed before
breakdowns. These appeared to be similar to breakdowns in that the current
rapidly pulses, but the amplitude of these pulses were orders of magnitude
lower.

The digitized signals from the anode current and SiPM were analyzed offline in
order to locate short pulses. The waveforms were processed in blocks of 106
samples and filtered using a zero-sum difference-of-Gaussians filter. Peaks in
the filtered response are treated as a pulse, with boundaries at the zero
crossings on either side of the peak. The particular widths of the filter were
tuned to the particular noise and electronics response of the run being
analyzed. A semi-regular chirp-like signal was present in background
acquisitions (see Fig. 8, top left), which was filtered out with a separate
algorithm. The signal is first filtered to find the standard deviation,
calculated over a fixed-length moving window. Regions where the ratio of the
signal amplitude to the standard deviation were sufficiently low were replaced
with a low-pass filter value. To further filter the chirp-like pulses, the
condition FWHM < $8.4\text{\,}\mathrm{\SIUnitSymbolMicro s}$ was applied.
Also, only anode pulses of negative polarity were selected. Several quantities
were calculated on each pulse and the slow control logs provided information
on the detector conditions during each pulse.

An algorithm to locate breakdowns in the DAQ was used to terminate each ramp
and group the pulses into an object for analysis. During a breakdown, a large
overshoot of the anode current signal was frequently seen, leading to what
appear to be large surges of positive current into the anode. These also
occasionally occurred in the seconds preceding breakdowns. Three such surges
in one event window were required to flag a breakdown. An example of a sub-
threshold discharge and a true breakdown event are shown in the bottom left
and bottom right of Fig. 8, respectively.

Figure 8: Various categories of event acquisitions. Top left: An example of
background noise. The oscillatory behaviour is removed through a filtering
technique. Top right: an example of the coincidence between anode and SiPM
pulses. This event was recorded close to the breakdown point. Bottom left: A
subthreshold discharge. These occasionally occur during a ramp, but did not
trigger an HVPS fault signal. Bottom right: A true breakdown event, coincident
with an HVPS fault signal.

The charge amplifier signal frequently saturated the dynamic range of the DAQ,
and large anode pulses were almost always associated with a corresponding
pulse in the SiPM channel, as shown in the last three plots in Fig. 8.
Typically, an increase in anode pulse rate of around a factor of 100 was
observed in the preceding 60 seconds to a breakdown. Due to the lower light
collection and high dark rate compared to the PMT, it was not possible to
observe a large amount of light in the SiPM until just a few
$\text{\,}\mathrm{m}\mathrm{s}$ before a breakdown. The anode pulses have a
distinctive spectrum of heights. Most pulses were barely above threshold, but
the overall distribution was bimodal, which was particularly evident when the
gain was lowered on the charge amplifier.

Figure 9:  _Top:_ Pulses found over the course of a single ramp. The anode and
SiPM pulses are nearly coincident with one another. The anode pulses have a
better signal-to-noise ratio, and therefore smaller peaks can be resolved on
this scale. _Bottom:_ Complementary information obtained from the slow control
logs for the same ramp. The burst in current at time 0 is not the breakdown
itself, but rather the discharge of current through the HV power supply due to
the sudden drop of electric potential energy.  Figure 10:  Distributions of
anode current preceding a breakdown for the following parameter scans in Run
10: electrode separation (top), pressure (middle), and ramp rate (bottom). The
anode current is averaged over in time bins of 1 second. Pulses within 2 event
windows (approximately 750 ms) of a subthreshold breakdown (defined in text)
were discarded. A general rise in anode current in the few tens of seconds
before breakdown is observed in all cases.

The anode current pulse rate was generally higher in runs with larger SEA,
while pressure and ramp speed had no significant effect on the time profile of
precursor pulses. The precursor rate within any particular ramp did not rise
continuously, but instead appeared to spike in rate several times preceding a
breakdown (see an example in Fig. 9). Overall, a consistent rise in anode
current before breakdown was observed. For instance, Fig. 10 shows the average
current (as measured by the charge amplifier) over time bins of
$1\text{\,}\mathrm{s}$ for the different parameter scans in Run 10. These
plots show a first surge in current around 50–$60\text{\,}\mathrm{s}$ and a
rapid increase in current in the last few seconds.

Comparing the polished and the passivated electrodes, we observed that
passivation only has a small effect on the preceding activity before a
breakdown. As Fig. 11 shows, there is only a moderate suppression in both the
anode current and SiPM rate in the immediate few seconds before a breakdowns
when comparing Run 9 (mechanically polished electrodes) to Run 10 (passivated
electrodes).

Figure 11: Comparison of anode current (top) and SiPM rate (bottom) in the 30
seconds preceding a breakdown between mechanically polished electrodes (Runs
9, orange line) and passivated electrodes (Run 10, blue line). The anode
current is averaged over in time bins of $1\text{\,}\mathrm{s}$. An electrode
separation of $2\text{\,}\mathrm{m}\mathrm{m}$ is considered in both cases.
The two sets of electrodes perform similarly, except for the last few seconds
in which the polished electrodes exhibits increased activity relative to the
passivated ones.

A conditioning effect over the course of a run was also observed. This is
illustrated in Fig. 12 for Runs 9 and 10. The current into the anode was
averaged over a time period well separated from either the breakdown itself or
the initial ramp. The region of $45\text{\,}\mathrm{s}$ to
$120\text{\,}\mathrm{s}$ from the start of the ramp was chosen to obtain a
large window and avoid capturing the immediate few seconds before a breakdown.
A downward trend is observed in both cases, indicating a reduction in charge
being emitted as more ramps were conducted during each run.

Figure 12:  Anode current as a function of HV ramp number for the entire Run 9
(top) and Run 10 (bottom), in which a fresh set of polished and passivated
electrodes were used, respectively. The anode current is integrated between
the first 45 and 120 seconds from the start of the HV ramp, with a minimum gap
of 20 seconds between the start and end of each ramp. A downward trend within
each run is observed in both cases, which can be attributed to conditioning of
the electrodes.

###  III.5 Position reconstruction

Videos were recorded during all runs in order to reconstruct the position of
breakdowns. Unfortunately, the number of videos acquired in Runs 5 through 8
was insufficient and some Run 9 videos were compromised by the field of view
of one of the cameras being blocked. From the remaining videos, the positions
of breakdowns were reconstructed: Run 9 at 1 mm and Run 10 data at 1, 2 and 3
mm electrode separation distances. The distributions of these breakdowns were
compared to the FN model of field emissions using simulations of the electric
field for the XeBrA geometry. For this comparison, the tilt between the
electrodes determined from the tilt estimation procedure, found in Table 1,
was used when simulating the electric field for a given Run. The resulting
electric field distribution was then used to find the corresponding FN surface
current density.

For each breakdown distribution, the Gaussian Kernel Density Estimate (KDE)
was found using the scikit-learn library Pedregosa  _et al._ (2011). The
centroids of the KDE and maximum values of the FN surface current were
recorded and are given in Table 5. The FN surface currents were rotated to
minimize the distance between the centroid of the KDE and the maximum of the
FN surface current. As an example, the subsequent FN surface current
densities, along with the breakdown distributions, are shown in Fig. 13 for
Run 9, 1 mm data.

Overall, the position distribution of breakdowns shows a direct relationship
with the simulated SEA. More specifically, broader SEA lead to broader
breakdown distributions. When these distributions are compared to the FN
surface current densities, related to probability of electron emission, the
distances between the centroids of the data and maximums of the FN currents
are, on average, found to be around 9 mm (see Table 5). These results are
favourable towards the FN model, but more detailed comparisons between
reconstructed positions and simulations are needed, which will be the object
of study of future XeBrA runs.

Figure 13: Comparison between breakdown distributions and FN current distributions from simulations for Run 9, 1 mm data. The breakdown distribution (points) are overlaid with the FN current densities (isocontours). Isocontours correspond to the 90, 80, 60, 30, and 10 $\%$ of the maximum FN current. The centroids of each distribution are marked with a red dot for data and a blue star for simulation, respectively. Run | Separation [mm] | N${}_{\text{rec}}$ | Centroid | Maximum | Distance [mm]
---|---|---|---|---|---
|  |  | of data | of FN |
|  |  | x, y [mm] | x, y [mm] |
9 | 1 | 47 | 10.2, 10.8 | 15.0, 15.7 | 6.9
10 | 1 | 46 | 3.9, 3.0 | 12.4, 9.7 | 10.8
10 | 2 | 50 | 5.0, 0.4 | 15.7, 1.3 | 10.8
10 | 3 | 50 | 9.1, 4.8 | 14.0, 7.4 | 5.6
Table 5: Centroids of both data and simulations for those datasets with the
largest number of reconstructed positions (N${}_{\text{rec}}$). The last
column shows the distance between the centroid of the reconstructed positions
and the maximum of the electric field simulation.

###  III.6 Visual observations

The presence of viewports oriented at right angles to the electrodes
facilitated visual observations of a variety of phenomena. Critically, this
provided certainty that HVPS current trips corresponded to breakdowns between
the electrodes as well as verifying the bubble-free state of the liquid inside
the chamber.

Breakdowns are associated with a distinct, bright arc and an audible cracking
sound (transmitted through the metal) at larger separation distances. Copious
boiling of LXe is usually followed after such event. More importantly,
evidence of some activity developing in the liquid around the breakdown
location prior to the main discharge was usually observed. These appear as
faint, flickering arcs between the electrodes. For instance, Fig. 14 shows an
example of two near-simultaneous breakdowns spawning from two visible arcs and
followed by two separate bubbles.

Figure 14: An example of a breakdown containing two apparent discharges. Four
frames are shown in sequential order from top to bottom. Two light spots
appear in the center of the frame before a big flash of light is emitted and
two bubbles emerge from that location immediately after.

An interesting possibility is that some of these early bright spots correspond
to bubbles that nucleated prior to a breakdown. This would corroborate the
hypothesis that bubbles are an essential mechanism for originating dielectric
breakdown. In most breakdowns, bubbles take a second or more to roll off the
side of the electrodes after a discharge and they also grow in size over time.
Therefore, it is unlikely that these bubbles floated in between the electrodes
from elsewhere in the apparatus. One particular good example of a bubble
possibly initiating a breakdown is shown in Fig. 15 (multimedia file). In this
short video, one can see a small bubble traveling from the anode to the
cathode on the left side of the frame and subsequently a breakdown occurs in
that same location.

Figure 15: Video of a bubble occurring prior to a breakdown. A small bubble
starts forming on the top left corner, travels down to the cathode’s surface
and a breakdown occurs in that same location (Multimedia view).

Perhaps one of the most intriguing visual observations was that of a
“shimmering” effect seen during some ramps. At large separations and during a
high voltage ramp, the LXe would begin to ripple, increasing in intensity
until a breakdown occurred, and then settling down to a more placid condition
in time for the next ramp. A close visual analogue for this effect would be
the appearance of mixing alcohol and water. It is worth noticing that this
effect was never observed with unbiased electrodes in stable conditions. We
hypothesize that this phenomenon was due to localized heating of the xenon,
leading to changes in the index of refraction. It is possible, as pointed out
in Ref. Atrazhev  _et al._ (2010), that such heating plays a key role in the
development of a discharge. If mixing is poor, the liquid may become locally
superheated until an instability occurs, leading to spontaneous nucleation of
bubbles, which then initiate a dielectric breakdown in the bulk of the
detector.

Finally, following several runs of XeBrA, it was noticed that some damage was
done to the electrodes in the form of pitting. The electrodes were examined
under a microscope to reveal pits of various sizes on both the cathode and
anode. Subsequent to this, the electrodes were replaced with duplicates and
the originals were re-polished. Following further pitting to the second pair
of electrodes similar to the first, separation distances of over 3 mm were
avoided to keep the stored energy lower. This avoided the more significant
pitting seen in the initial set of electrodes, although some damage was still
done over the course of a run.

##  IV  Conclusion

With the XeBrA experiment, which employs a pair of Rogowski electrodes with
adjustable separation distance, we confirmed that breakdown field scales
inversely with stressed electrode area in LXe for stressed areas up to 33 cm2.
No significant, reproducible scaling with LXe pressure was observed between
1.5 bar and 2.2 bar absolute. Similarly, no significant dependence was
observed with the high voltage ramp speed in the range between
$100\text{\,}\mathrm{V}\mathrm{/}\mathrm{s}$ and
$200\text{\,}\mathrm{V}\mathrm{/}\mathrm{s}$. By contrast, a small increase in
breakdown field was observed with the passivation of the electrode surfaces.

The location of a breakdown on the electrode surface was reconstructed from
high-speed videos. They show moderate correlations with FN current as obtained
by electric field simulations. Moreover, evidence for bubble nucleation
initiating breakdown was observed with low-speed videos. We hypothesize that
the bubble nucleation itself is caused by localized heating due to field
emission from field enhancers, such as local asperities.

Finally, some practical recommendations can be drawn from this work. First,
significant breakdown precursor activity was observed. Small discharges were
detected with a SiPM and a charge-sensitive amplifier connected to the anode.
These discharges slowly increase in intensity in the last 30 seconds before a
breakdown and sharply accelerate in the last second. Hence, having the ability
to ramp down the voltage on the scale of seconds could be beneficial for
avoiding electrical breakdown. Second, a downward trend in current collected
at the anode was observed over the course of a run, which motivates conducting
an initial conditioning campaign prior to the start of any science data-taking
campaign. Third, the extrapolation from our measured stressed area scaling
shows that the breakdown field for the next-generation, LXe TPC experiments,
will only be on the order of a few tens of
$\text{\,}\mathrm{k}\mathrm{V}\mathrm{/}\mathrm{c}\mathrm{m}$. These results
have important implications for the design of the high voltage delivery system
of such experiments.

###### Acknowledgements.

We are grateful to the support received from the technical and engineering
staff at LBNL. Also, we thank Takeyasu Ito and Nguyen Phan for helpful
correspondence. Funding for this work is supported by the U.S. Department of
Energy, Office of Science, Office of High Energy Physics under Contract
Numbers DE-SC0018982 and DE-AC02-05CH11231.

## Data Availability Statement

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

##  Appendix A Plumbing diagram of XeBrA

See Fig. 16.

Figure 16: Piping and instrumentation (P&ID) of the XeBrA experiment.

##  Appendix B Summary of fit results

See Table 6

Dataset | Electrodes | Tilt | Gap | Ramp Rate | Pressure | N | Model | k | $E_{0}$ | $E_{1}$ | $E_{2}$
---|---|---|---|---|---|---|---|---|---|---|---
|  | [$\circ$] | [mm] | [V/s] | [bar] |  |  |  | [kV/cm] | [kV/cm] | [kV/cm]
400 | MP1 | 0.61 | 3.0 | 100 | 1.70 | 19 | 3 | 3.1 $\pm$ 2.3 | 2.8 $\pm$ 1.9 | 28.6 $\pm$ 1.8 | 31.4 $\pm$ 0.2
401 | MP1 | 0.61 | 1.0 | 100 | 2.00 | 66 | 3 | 5.6 $\pm$ 3.8 | 37.5 $\pm$ 22.2 | 77.4 $\pm$ 21.3 | 114.9 $\pm$ 1.3
402 | MP1 | 0.61 | 2.0 | 100 | 2.00 | 60 | 3 | 16.4 $\pm$ 12.3 | 78.4 $\pm$ 55.4 | 6.3 $\pm$ 54.9 | 84.7 $\pm$ 0.7
403 | MP1 | 0.61 | 3.0 | 100 | 2.00 | 145 | 3 | 12.2 $\pm$ 0.0 | 77.1 $\pm$ 0.0 | 0.0 $\pm$ 0.0 | 77.1 $\pm$ 0.0
404 | MP1 | 0.61 | 5.0 | 100 | 2.00 | 40 | 1 | 15.4 $\pm$ 1.9 | 71.1 $\pm$ 0.7 | 0.0 $\pm$ 0.0 | 71.1 $\pm$ 0.7
500 | MP1 | 0.61 | 1.0 | 100 | 2.01 | 35 | 3 | 44.7 $\pm$ 8.1 | 83.5 $\pm$ 13.0 | 4.9 $\pm$ 13.0 | 88.4 $\pm$ 0.2
501 | MP1 | 0.61 | 2.0 | 100 | 2.00 | 81 | 3 | 12.1 $\pm$ 2.8 | 82.4 $\pm$ 1.7 | 0.0 $\pm$ 0.0 | 82.4 $\pm$ 1.7
504 | MP1 | 0.61 | 2.0 | 100 | 2.00 | 20 | 1 | 9.8 $\pm$ 1.7 | 61.0 $\pm$ 1.4 | 0.0 $\pm$ 0.0 | 61.0 $\pm$ 1.4
505 | MP1 | 0.61 | 3.0 | 100 | 2.00 | 34 | 2 | 1.5 $\pm$ 0.2 | 9.4 $\pm$ 1.3 | 41.5 $\pm$ 0.3 | 50.9 $\pm$ 1.2
506 | MP1 | 0.61 | 5.0 | 100 | 1.98 | 97 | 3 | 9.4 $\pm$ 12.6 | 29.0 $\pm$ 46.6 | 3.0 $\pm$ 15.6 | 32.0 $\pm$ 56.0
601 | MP1 | 0.28 | 1.0 | 100 | 2.01 | 53 | 3 | 1.9 $\pm$ 0.4 | 20.2 $\pm$ 3.3 | 81.7 $\pm$ 1.6 | 101.8 $\pm$ 1.8
602 | MP1 | 0.28 | 2.3 | 100 | 2.01 | 21 | 2 | 1.8 $\pm$ 0.2 | 14.5 $\pm$ 1.9 | 41.0 $\pm$ 0.5 | 55.5 $\pm$ 1.5
603 | MP1 | 0.28 | 2.0 | 100 | 2.01 | 38 | 1 | 7.8 $\pm$ 1.0 | 77.2 $\pm$ 1.7 | 0.0 $\pm$ 0.0 | 77.2 $\pm$ 1.7
701 | MP1 | 0.28 | 5.0 | 100 | 2.00 | 42 | 3 | 9.8 $\pm$ 16.3 | 53.1 $\pm$ 118.3 | 19.1 $\pm$ 119.9 | 72.1 $\pm$ 4.5
703 | MP1 | 0.28 | 1.0 | 100 | 2.00 | 50 | 1 | 11.9 $\pm$ 1.4 | 111.9 $\pm$ 1.3 | 0.0 $\pm$ 0.0 | 111.9 $\pm$ 1.3
704 | MP1 | 0.28 | 2.0 | 100 | 2.00 | 37 | 1 | 17.0 $\pm$ 2.2 | 91.5 $\pm$ 0.9 | 0.0 $\pm$ 0.0 | 91.5 $\pm$ 0.9
705 | MP1 | 0.28 | 3.0 | 100 | 2.00 | 40 | 3 | 19.2 $\pm$ 2.7 | 85.7 $\pm$ 3.9 | 0.3 $\pm$ 2.9 | 86.0 $\pm$ 1.2
706 | MP1 | 0.28 | 2.0 | 100 | 1.70 | 41 | 1 | 17.0 $\pm$ 2.1 | 90.9 $\pm$ 0.8 | 0.0 $\pm$ 0.0 | 90.9 $\pm$ 0.8
707 | MP1 | 0.28 | 2.0 | 100 | 2.20 | 44 | 1 | 18.0 $\pm$ 2.1 | 97.9 $\pm$ 0.8 | 0.0 $\pm$ 0.0 | 97.9 $\pm$ 0.8
803 | MP2 | 1.04 | 3.0 | 100 | 2.00 | 47 | 1 | 10.2 $\pm$ 1.2 | 79.2 $\pm$ 1.2 | 0.0 $\pm$ 0.0 | 79.2 $\pm$ 1.2
804 | MP2 | 1.04 | 2.0 | 100 | 2.00 | 36 | 1 | 18.4 $\pm$ 2.3 | 75.3 $\pm$ 0.7 | 0.0 $\pm$ 0.0 | 75.3 $\pm$ 0.7
805 | MP2 | 1.04 | 1.0 | 100 | 2.00 | 36 | 3 | 34.8 $\pm$ 3.9 | 78.4 $\pm$ 3.2 | 0.6 $\pm$ 3.2 | 79.0 $\pm$ 0.3
806 | MP2 | 1.04 | 3.0 | 100 | 2.00 | 18 | 1 | 12.0 $\pm$ 2.0 | 67.2 $\pm$ 1.4 | 0.0 $\pm$ 0.0 | 67.2 $\pm$ 1.4
807 | MP2 | 1.04 | 2.0 | 100 | 2.00 | 18 | 3 | 25.4 $\pm$ 22.9 | 80.3 $\pm$ 1.9 | 0.0 $\pm$ 1.7 | 80.3 $\pm$ 1.8
808 | MP2 | 1.04 | 1.0 | 100 | 2.00 | 33 | 2 | 1.6 $\pm$ 0.1 | 3.6 $\pm$ 0.3 | 69.2 $\pm$ 0.1 | 72.8 $\pm$ 0.3
809 | MP2 | 1.04 | 2.0 | 100 | 1.50 | 42 | 3 | 3.6 $\pm$ 0.7 | 7.8 $\pm$ 1.4 | 72.5 $\pm$ 1.0 | 80.4 $\pm$ 0.5
810 | MP2 | 1.04 | 2.0 | 100 | 1.60 | 37 | 3 | 51.1 $\pm$ 23.3 | 66.7 $\pm$ 27.6 | 10.9 $\pm$ 27.5 | 77.6 $\pm$ 0.3
811 | MP2 | 1.04 | 2.0 | 100 | 1.70 | 34 | 3 | 2.6 $\pm$ 0.6 | 6.6 $\pm$ 0.9 | 71.9 $\pm$ 0.5 | 78.5 $\pm$ 0.6
812 | MP2 | 1.04 | 2.0 | 100 | 1.80 | 38 | 3 | 19.5 $\pm$ 9.6 | 59.9 $\pm$ 26.6 | 17.9 $\pm$ 26.4 | 77.8 $\pm$ 0.9
813 | MP2 | 1.04 | 5.0 | 100 | 2.00 | 39 | 1 | 13.3 $\pm$ 1.6 | 83.6 $\pm$ 1.0 | 0.0 $\pm$ 0.0 | 83.6 $\pm$ 1.0
900 | MP1 | 1.26 | 3.0 | 100 | 2.00 | 29 | 1 | 13.2 $\pm$ 2.0 | 86.5 $\pm$ 1.2 | 0.0 $\pm$ 0.0 | 86.5 $\pm$ 1.2
901 | MP1 | 1.26 | 3.0 | 100 | 2.00 | 47 | 1 | 13.1 $\pm$ 1.4 | 96.5 $\pm$ 1.1 | 0.0 $\pm$ 0.0 | 96.5 $\pm$ 1.1
902 | MP1 | 1.26 | 1.0 | 100 | 2.00 | 67 | 3 | 10.6 $\pm$ 4.0 | 134.0 $\pm$ 4.5 | 0.0 $\pm$ 0.0 | 134.0 $\pm$ 4.5
903 | MP1 | 1.26 | 2.0 | 100 | 2.00 | 81 | 1 | 10.1 $\pm$ 0.9 | 96.2 $\pm$ 1.1 | 0.0 $\pm$ 0.0 | 96.2 $\pm$ 1.1
904 | MP1 | 1.26 | 2.0 | 150 | 2.00 | 33 | 1 | 16.0 $\pm$ 2.0 | 106.2 $\pm$ 1.2 | 0.0 $\pm$ 0.0 | 106.2 $\pm$ 1.2
905 | MP1 | 1.26 | 2.0 | 200 | 2.00 | 35 | 1 | 10.2 $\pm$ 1.3 | 98.1 $\pm$ 1.7 | 0.0 $\pm$ 0.0 | 98.1 $\pm$ 1.7
906 | MP1 | 1.26 | 2.0 | 100 | 1.70 | 34 | 1 | 13.7 $\pm$ 1.7 | 107.9 $\pm$ 1.4 | 0.0 $\pm$ 0.0 | 107.9 $\pm$ 1.4
907 | MP1 | 1.26 | 2.0 | 100 | 1.50 | 40 | 1 | 13.8 $\pm$ 1.6 | 104.0 $\pm$ 1.2 | 0.0 $\pm$ 0.0 | 104.0 $\pm$ 1.2
1000 | Pass. | 0.51 | 3.0 | 100 | 2.00 | 19 | 1 | 14.1 $\pm$ 2.4 | 78.9 $\pm$ 1.3 | 0.0 $\pm$ 0.0 | 78.9 $\pm$ 1.3
1001 | Pass. | 0.51 | 3.0 | 100 | 2.00 | 40 | 2 | 2.2 $\pm$ 0.4 | 28.6 $\pm$ 3.7 | 46.2 $\pm$ 1.9 | 74.8 $\pm$ 2.3
1002 | Pass. | 0.51 | 1.0 | 100 | 2.00 | 71 | 3 | 15.4 $\pm$ 2.5 | 148.6 $\pm$ 2.3 | 0.0 $\pm$ 0.1 | 148.6 $\pm$ 2.3
1003 | Pass. | 0.51 | 2.0 | 100 | 2.00 | 78 | 3 | 8.6 $\pm$ 3.8 | 103.6 $\pm$ 6.1 | 0.0 $\pm$ 0.0 | 103.6 $\pm$ 6.1
1004 | Pass. | 0.51 | 2.0 | 150 | 2.00 | 48 | 3 | 13.8 $\pm$ 11.3 | 113.5 $\pm$ 5.1 | 0.0 $\pm$ 0.0 | 113.5 $\pm$ 5.1
1005 | Pass. | 0.51 | 2.0 | 200 | 2.00 | 38 | 1 | 12.3 $\pm$ 1.6 | 117.5 $\pm$ 1.6 | 0.0 $\pm$ 0.0 | 117.5 $\pm$ 1.6
1006 | Pass. | 0.51 | 2.0 | 100 | 1.60 | 42 | 1 | 12.8 $\pm$ 1.5 | 102.0 $\pm$ 1.3 | 0.0 $\pm$ 0.0 | 102.0 $\pm$ 1.3
1007 | Pass. | 0.51 | 2.0 | 100 | 1.50 | 40 | 1 | 15.2 $\pm$ 1.8 | 117.0 $\pm$ 1.2 | 0.0 $\pm$ 0.0 | 117.0 $\pm$ 1.2
1008 | Pass. | 0.51 | 2.0 | 100 | 1.70 | 42 | 1 | 13.9 $\pm$ 1.7 | 122.7 $\pm$ 1.4 | 0.0 $\pm$ 0.0 | 122.7 $\pm$ 1.4
Table 6: The results of the Weibull fitting procedure on a per-dataset basis.
The last two digits of the first column indicates the dataset number and the
number in front represents the run number. Three sets of electrodes were
employed: two mechanically polished (MP1 and MP2) and one with passivated
surfaces (Pass.). The cathode tilt angle, electrode gap separation, ramp rate,
and pressure for each dataset are also indicated. Column “N” shows the number
of breakdown events contained in each dataset after applying data quality cuts
and column “Model” indicates the model that was selected to fit the breakdown
field data (following the procedure described in Section III.1). Datasets with
less than 15 breakdown events were discarded, amounting to 10 in total. 7
additional datasets were discarded because of unfavorable conditions were
found with the experimental setup after the start of data acquisition. The
best-fit for the Weibull model parameters (defined in Section I.2) are shown
in the last 4 columns.

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