# A study of DC electrical breakdown in liquid helium through analysis of the
empirical breakdown field distributions

N. S. Phan<EMAIL_ADDRESS>Los Alamos National Laboratory, Los Alamos, New
Mexico 87545, USA  W. Wei  Los Alamos National Laboratory, Los Alamos, New
Mexico 87545, USA  B. Beaumont  Department of Physics, North Carolina State
University, Raleigh, NC 27695  N. Bouman  Department of Physics and Astronomy,
Valparaiso University, Valparaiso, IN 46383  S. M. Clayton  Los Alamos
National Laboratory, Los Alamos, New Mexico 87545, USA  S. A. Currie  Los
Alamos National Laboratory, Los Alamos, New Mexico 87545, USA  T. M. Ito
<EMAIL_ADDRESS>Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
J. C. Ramsey  Los Alamos National Laboratory, Los Alamos, New Mexico 87545,
USA  G. M. Seidel<EMAIL_ADDRESS>Department of Physics, Brown
University, Providence, Rhode Island 02912

###### Abstract

We report results from a study on electrical breakdown in liquid helium using
near-uniform-field stainless steel electrodes with a stressed area of
$\sim$0.725 cm2. The distribution of the breakdown field is obtained for
temperatures between 1.7 K and 4.0 K, pressures between the saturated vapor
pressure and 626 Torr, and with electrodes of different surface polishes. A
data-based approach for determining the electrode-surface-area scaling of the
breakdown field is presented. The dependence of the breakdown probability on
the field strength as extracted from the breakdown field distribution data is
used to show that breakdown is a surface phenomenon closely correlated with
Fowler-Nordheim field emission from asperities on the cathode. We show that
the results from this analysis provides an explanation for the supposed
electrode gap-size effect and also allows for a determination of the
breakdown-field distribution for arbitrary shaped electrodes. Most
importantly, the analysis method presented in this work can be extended to
other noble liquids to explore the dependencies for electrical breakdown in
those media.

##  I Introduction

There has been a long-standing interest in electrical breakdown in liquid
helium (LHe), motivated by practical applications as well as fundamental
interest. LHe may be used as electric insulator and thermal conductor at
cryogenic temperatures, which may simplify the design of superconducting
magnets[1]. Studying the behavior of LHe under a strong electric field may be
of interest in understanding electronic properties of condensed helium[2].
More recently, the study of electrical breakdown of LHe is motivated by the
importance of superfluid LHe as a medium in which experiments in nuclear,
particle, and astroparticle physics are performed, along with other noble
liquids, in particular LAr and LXe.[3] For some experiments, the application
of a high electric field is necessary. These include experiments to search for
the electric dipole moment of the neutron[4, 5, 6, 7, 8] and experiments to
look for light dark matter particles.[9, 10, 11, 12] For these experiments,
breakdown of LHe in response to DC electric fields (as opposed to AC or pulsed
electric fields) is particularly relevant. For a new method to generate a
large electric potential ($\sim$1 MV) inside LHe, see Ref. 13.

The study of electrical breakdown in LHe dates back to the late 1950s. [14,
15] Since then, active investigations have been conducted by many authors.
These studies employed various electrode geometries, electrode materials, and
temperature and pressure of LHe. The electrode geometries employed include
sphere to sphere, sphere to plane, and uniform field electrodes. The materials
used include stainless steel, tungsten, brass, copper, bronze, lead, niobium,
and others. Data exist for the temperature range of $1.2-4.2$ K mostly at the
saturated vapor pressure, with the bulk of the data taken at 4.2 K111Ref. 7
reports measurements at 0.4 K. A review of earlier work can be found in e.g.
Ref. 1. While the existing data show little consistency in general, several
parameters have been identified as potentially affecting breakdown, including
(1) electrode material and surface finish, (2) gap size and electrode size,
and (3) temperature and pressure of the liquid.

Based on a detailed theoretical study of electron multiplication induced by an
electric field in LHe starting from the kinetic Boltzmann equation [16], the
intrinsic breakdown field of LHe is expected to be at least in the low MV/cm
region. On the other hand, breakdown fields around or below a few 100’s kV/cm
have been commonly reported in the literature for LHe. These observations are
therefore presumed to be the result of the following process: (1) Field
emission of electrons occurs on a rough surface because of extremely high
local fields at sharp asperities. (2) This causes local heating, which in turn
results in vapor bubble formation leading to electron multiplication and vapor
growth proceeding together. (3) Breakdown is then the result of a vapor column
extending to the anode. Note that in other dielectric liquids, formation and
growth of vapor bubbles associated with electrical breakdown have been
observed experimentally[17, 18, 19].

This picture is at least qualitatively consistent with the observed dependence
of the breakdown field on the electrode material and surface finish, the
electrode size, and the pressure of liquid helium. However, there is no model
that can quantitatively describe the dependencies of the breakdown field on
these parameters.

The study reported in this paper was performed as part of the research and
development (R&D) for the SNS nEDM experiment.[20, 21, 8] This experiment,
currently under development to be mounted at the Spallation Neutron Source
(SNS) at the Oak Ridge National Laboratory, will search for the permanent
electric dipole moment of the neutron (nEDM) using a method proposed by Golub
and Lamoreaux[4]. The experiment will be performed inside a bath of LHe at
approximately 0.4 K, and large-area electrodes (61 cm $\times$ 30 cm) will
provide an electric field of up to $\sim$75 kV/cm inside the experimental
volume. Because the sensitivity of the experiment is directly proportional to
the strength of this applied field, it is of critical importance to establish
the following: (1) the highest electric field that can be sustained stably;
(2) how the breakdown depends on factors such as the pressure and temperature
of LHe, the electrode material and surface condition, and the size of the
system (the electrode area and/or the size of the gap between electrodes); and
(3) what distribution describes the breakdown field (or voltage).

However, studying all these aspects of electrical breakdown in LHe using a
full-scale SNS nEDM system would be a rather daunting task, considering that
it would require 1500 l of LHe to be cooled to $\sim$ 0.4 K for each set of
measurements requiring a change in the cryogenic apparatus. While the ultimate
performance test of the electrodes to be used for the SNS nEDM experiment will
need to be conducted utilizing a full scale system, it is possible to obtain
information on the dependence of the breakdown properties on the electrode
materials and LHe temperature and pressure using a much smaller system, which
is significantly easier to operate. In addition, since the breakdown is a
stochastic phenomenon, how the breakdown properties scale with the size of the
system depends on and can be inferred from the breakdown-field distribution
for a small system, as discussed in this paper. Therefore significant insights
can be gained from a detailed study using a small-scale system.

To this end, we have performed a comprehensive study of electrical breakdown
in LHe using an apparatus with small electrodes. We refer to this apparatus as
the “Small Scale High Voltage Test Apparatus” (SSHV apparatus), which can
accommodate electrodes with a high field area of $\sim$0.725 cm2. Using this
apparatus, we studied (1) the dependence of the breakdown field on the LHe
pressure, LHe temperature, and electrode surface finish, (2) the distribution
of the breakdown field with a constant field ramp rate, and (3) distribution
of the time it takes before a breakdown occurs when a field is set at some
predetermined value. The dependence of the probability of breakdown on the
field strength, extracted from the data, closely resembles that of the field
emission, giving a strong indication that the initial process involves the
field emission from the cathode. Also, having the dependence of the
probability of breakdown on the field strength determined for one electrode
size naturally allowed us to predict how the breakdown field scales with the
surface area of the electrode. Our prediction agrees well with data taken with
our “Large Scale High Voltage Test Apparatus” (LSHV apparatus) with an
electrode area of $\sim$1100 cm2, which had previously been constructed in our
laboratory[22] as well as those obtained by other investigators. In addition,
our study shows that what was thought be the gap size dependence can be
explained by the surface area scaling when the change in the stressed area
caused by the change in the gap size is properly taken into consideration.

In this paper, we describe the SSHV apparatus, the measurements performed
using it, the analysis of the data, and the above-mentioned findings and their
implications, as well as our findings on the dependencies of the breakdown
field and distributions on various parameters.

The paper is organized as follows. In Sec. II we describe the design of the
SSHV apparatus. In addition, we give a brief description of the LSHV system.
We describe the experimental procedure and present the data in Sec. III. Our
analysis of the data is presented in Sec. IV, where we also discuss the
implications of our findings.

##  II Apparatuses

###  II.1 Small scale high voltage test apparatus

The design of the SSHV apparatus was guided by the following requirements: (1)
the electrodes have a finite uniform field region with good electrode
alignment, (2) the temperature can be varied between 1.7 K and 4.0 K, (3) the
pressure can be varied, (4) the pressure can be measured reliably, and (4) the
apparatus can fit in the existing Dewar (Janis Model 10 CNDT Research
Dewar[23]).

Figure 1: Schematics of the SSHV system. (a) Flow diagram, showing the Dewar
LHe volume, CV, various lines, and the high voltage (HV) feed line. (b) Scale
drawing of the SSHV insert along with expanded view of the CV  Figure 2:
Schematic of the SSHV electrode assembly

Schematic diagrams of the apparatus are shown in Fig. 1. The electrodes were
supported by a structure made of polyamide-imide, which has a similar
coefficient of thermal expansion to stainless steel, the electrode material,
and is a good electrical insulator (see Fig. 2). The electrode system was then
placed inside the “Central Volume” (CV) enclosure, made of stainless steel.
The CV was inserted into the LHe volume of the Janis Dewar. LHe was introduced
into the CV through the CV fill valve. The CV was cooled by evaporatively
cooling the LHe outside the CV. Leaving the CV fill valve open ensured that
the pressure of the LHe inside the CV was at the saturated vapor pressure
(SVP). When the LHe inside the CV needed to be pressurized to a pressure
higher than the SVP, the CV fill valve was closed. The CV was connected to an
external source of high purity helium gas through a small diameter tubing. The
pressure of the LHe inside the CV was controlled by adjusting the pressure of
the external helium gas source with the CV fill valve closed. In this way, the
temperature ($T$) and pressure ($P$) were varied independently in the ranges
of $1.7<T<4.0$ K and SVP $<P<626$ Torr.

The high voltage (HV) feed line was designed with the following three design
considerations in mind:

(1) The heat transmitted to the HV electrode through the HV feed line needs to
be minimized because a small amount of heat imparted to the HV electrode can
boil LHe on the electrode surface due to the rather small heat of vaporization
of LHe, creating bubbles and significantly reducing the breakdown field.

(2) The conductor carrying HV should not be directly exposed to the volume
above the LHe surface, which is filled with helium vapor, as it would cause
electrical breakdown at relatively low voltages.

(3) The capacitance of the HV feed line needs to be minimized. Electrical
breakdown causes the energy stored capacitively in the HV feed line to be
released, significantly altering the surface condition of the electrodes or
even damaging the electrodes.

In order to meet these considerations, the HV was brought in from the outside
of the cryostat to the inside of the CV using a feed line that consisted of a
stainless steel wire placed inside an evacuated tube. Using vacuum as
electrical insulation (instead of dielectric) kept the capacitance low. At the
same time, it prevented the HV conductor from being exposed to the helium
vapor above the LHe surface. At the ends of the evacuated tube were 40-kV
ceramic HV feedthroughs, providing seals between the atmosphere and the vacuum
inside the tube on the room temperature end, and the vacuum inside the tube
and the LHe inside the CV on the other end. The HV feedthrough on the cold end
was mounted on the top wall of the CV, providing thermal anchoring to the HV
wire. The wire diameter was chosen to minimize the heat flow from the room
temperature while meeting the electrostatic requirement that the field on the
surface of the wire was sufficiently low when the full 40 kV was applied. The
heat through the feed tube wall was removed by the LHe outside the CV, as the
tube wall was directly in contact with the LHe.

The electrodes were made of stainless steel. The shape of the electrodes was
chosen so that the electric field on the surface of each electrode was near
uniform in a region around the center and that the field rapidly decreased
outside this region. Such an arrangement simplified the data analysis reported
in Sec. IV.

A field distribution, calculated using the finite element method[24], is shown
in Fig. 3 for both the HV and ground electrode for a potential difference of
10 kV and a gap distance of 0.44 mm. We measured the average gap distance to
be 0.52 mm at room temperature. Based on the known coefficients of thermal
expansion for the materials used for the electrode assembly, we estimate that
the gap distance shrinks by 15% when the system is cooled, giving an average
gap distance of 0.44 mm. The estimated overall uncertainty on this quantity is
20% due to variation in electrode thicknesses.

When the gap was set to be 2 mm and the pressure was set to be 600 Torr, no
breakdown was observed even at 40 kV. This confirms that all the HV components
were able to withstand potentials up to 40 kV and that the breakdowns observed
were indeed in the gap between the two electrodes.

Figure 3: Calculated distribution of the electric field strength on (a) the HV
electrode and (b) the ground electrode for a potential difference of 10 kV and
a gap distance of 0.44 mm.

The HV power supply for the SSHV system was computer controlled. The ground
electrode was connected to a trip circuit, which shuts off the HV power supply
as soon as it detected a current flowing out of the ground electrode. The
circuit had a comparator that tripped and latched when the current exceeded
100 $\mu$A with the 1-k$\Omega$ load resistor. This shut off a relay switch to
the interlock of the power supply and caused the stoppage of the voltage ramp.
The breakdown voltage was then recorded and a wait time of a few seconds
passed before the latched interlock was reset. Then a new voltage ramp
sequence was started. Refer to Fig. 4.

Figure 4: A schematic showing electrical connections for the SSHV system.

###  II.2 Large scale high voltage test apparatus

A schematic of the LSHV apparatus is shown in Fig. 5. Details of its design
and operation can be found in Ref. 22. Here we give a brief description of the
apparatus relevant for the measurements reported in this paper.

The CV (= Central Volume, a volume filled with LHe) was a cylinder with an
inner diameter of 66 cm and an inner length of 44 cm. The CV housed the HV,
ground, and charging electrodes. Both the HV and ground electrodes had a
diameter of 45.8 cm. However, since the HV electrode had a thickness of 10.2
cm and the edge was rounded with a radius of curvature of 5.1 cm, the flat
area had a diameter of 35.6 cm. Using the finite element method [24] we
calculated the effective stress area to be 1,100 cm2. The ground electrode was
designed to be movable and the charging electrode was designed to be
disconnectable from the HV electrode, because one of the purposes of this
apparatus was to demonstrate the concept of HV amplification based on a
variable capacitor as described in Ref. 22. However, for acquiring the data
reported in this paper, these features were not used. The position of the
ground electrode was fixed to 3 mm and the charging electrode remained
connected to the HV electrode.

The temperature and pressure of the LHe were varied by directly pumping on the
LHe in the CV. Because of this arrangement, the pressure was always the
saturated vapor pressure at a given temperature. In order to avoid the
bubbling of LHe affecting the breakdown field measurements, the measurements
were made with the pumping turned off and the volume closed off after the
desired pressure and temperature were reached.

Figure 5: A schematic of the LSHV CV (Central Volume, the volume filled with
LHe).

##  III Experimental procedure and results

###  III.1 Procedure

Data were acquired for three different electrode surface polish configurations
under varying pressures and temperatures. These configurations were:
electropolished (EP), mechanically-polished (MP) and hybrid-polish (HP), where
the latter configuration consisted of an electropolished electrode in
conjunction with a mechanically-polished electrode. For a given set of
operating conditions (temperature, pressure, and electrode polish), the
voltage on the HV electrode was ramped up from zero at a constant rate until a
breakdown occurred and which caused the power supply to trip (or in the case
of no breakdown until 40 kV was reached). The voltage at which the breakdown
was observed and the operating conditions were recorded. This procedure was
repeated many times to obtain a set of breakdown voltages for the given
operating conditions. The parameters and conditions under which the datasets
were acquired are summarized in Tabs. 1 and 2 for the SSHV and LSHV
apparatuses, respectively. Henceforth, for brevity a dataset will be
referenced by the polish of the electrodes used in the measurement and its set
number (e.g., EP:3, HP:24, or MP:32).

The majority of the datasets were acquired at both fixed temperature and
pressure and with a constant ramp rate, but several datasets were acquired
under slightly different procedure and conditions. Dataset EP:1 was acquired
at a fixed temperature of 1.92 K but with variable pressure. Dataset MP:50 was
acquired at a fixed pressure of 480 Torr but with the temperature increasing
from $2.2-4.0$ K as the system was allowed to warm up. The same procedure was
applied to acquire dataset MP:38. Dataset MP:53 was acquired at a fixed
temperature and pressure but with three different voltage ramp rates of 50
V/s, 100 V/s, and 200 V/s. The dependence of the breakdown on the ramp rate is
discussed in further details in Sec. III.2.2. Finally, datasets $55-58$
involved measurements of the time to breakdown and the procedure used in
acquiring these data are discussed in more detail in Sec. IV.

###  III.2 Results

####  III.2.1 Distribution of breakdown field

The breakdown voltages were converted to breakdown fields through, $E=V/d$,
where $d$ was the separation between the HV and ground electrodes and $V$ was
the voltage at which a breakdown was observed. The uncertainty on this
distance is estimated to be 20% for the SSHV data and 15% for the LSHV data.
The breakdown field distribution for selected SSHV datasets acquired with the
electropolished, mechanically-polished, and hybrid electrodes are shown in
Fig. 6. The distributions for the LSHV datasets are not shown because each
dataset contained only a few breakdown measurements.

From Fig. 6, the mean values of the distributions are seen to be shifted to
higher fields with increasing pressure for all types of electrode polish.
Interestingly, the widths of the electropolished distributions are much
broader than the distributions obtained with the other electrode polishes. We
will discuss a possible explanation for this observation in Sec. IV.
Furthermore, a quite intriguing feature is the absence of any significant
differences between the distributions acquired with mechanically-polished
electrodes as compared to those acquired with the hybrid electrodes. This has
important implications regarding whether breakdown is initiated on the cathode
or anode surface. We will discuss this in more detail in Sec. IV.

Table 1: SSHV datasets and the operational conditions in which they were acquired. The surface polish of the electrodes is specified by electropolished(EP), hybrid-polish(HP), and mechanically-polished(MP). Set | Electrode polish | No. breakdowns | P (Torr) | T (K) | Ramp rate (V/s) |  $\mkern 1.5mu\overline{\mkern-1.5muE\mkern-1.5mu}\mkern 1.5mu$ (kV/cm) |  $\sigma_{\text{br}}$ (kV/cm)
---|---|---|---|---|---|---|---
1777Measurements made during cooling with $P$ between 12.6 - 35.2 Torr at SVP. | EP | 104 | 19.0 | 1.93 | 100 | 349 | 83
2 | EP | 84 | 12.6 | 1.81 | 100 | 415 | 121
3 | EP | 93 | 12.7 | 1.81 | 100 | 436 | 124
4 | EP | 86 | 12.9 | 1.81 | 100 | 473 | 127
5 | EP | 91 | 12.9 | 1.81 | 100 | 430 | 139
6 | EP | 109 | 7.4 | 1.67 | 100 | 425 | 134
11 | MP | 80 | 8.8 | 1.75 | 100 | 213 | 47
12 | MP | 81 | 193.5 | 3.01 | 100 | 310 | 51
13 | MP | 82 | 402.0 | 3.61 | 100 | 414 | 66
14 | MP | 61 | 617.2 | 4.01 | 100 | 525 | 86
15 | MP | 40 | 621.4 | 4.01 | 100 | 499 | 82
16 | MP | 40 | 193.4 | 3.05 | 100 | 365 | 56
17 | MP | 40 | 9.0 | 1.72 | 100 | 253 | 41
21 | HP | 80 | 9.6 | 1.73 | 100 | 222 | 50
22 | HP | 80 | 202.9 | 3.08 | 100 | 316 | 50
23 | HP | 81 | 395.9 | 3.60 | 100 | 358 | 55
24 | HP | 82 | 618.6 | 4.01 | 100 | 425 | 54
25 | HP | 38 | 8.0 | 1.69 | 100 | 242 | 63
26 | HP | 30 | 469.5 | 3.75 | 100 | 442 | 81
27 | HP | 83 | 617.5 | 4.01 | 100 | 479 | 56
31 | MP | 100 | 9.1 | 1.72 | 100 | 162 | 32
32 | MP | 83 | 330.2 | 1.75 | 100 | 399 | 31
33 | MP | 135 | 324.1 | 3.43 | 100 | 353 | 46
34 | MP | 80 | 9.8 | 1.74 | 100 | 191 | 41
35141414Negative polarity on HV electrode. | MP | 105 | 9.8 | 1.74 | 100 | 196 | 34
36 | MP | 107 | 606.9 | 1.76 | 100 | 495 | 32
37 | MP | 123 | 625.9 | 4.02 | 100 | 438 | 56
38161616$P$ fixed with $T$ increasing from 2.2 to 2.9 K as a natural warm-up. | MP | 40 | 202.2 | 2.56 | 100 | 370 | 41
39 | MP | 135 | 131.0 | 1.76 | 100 | 370 | 51
40 | MP | 117 | 140.5 | 2.84 | 100 | 305 | 46
41 | MP | 116 | 10.5 | 1.76 | 100 | 211 | 50
42 | MP | 120 | 484.2 | 1.77 | 100 | 507 | 53
43 | MP | 118 | 487.1 | 3.78 | 100 | 427 | 69
44 | MP | 91 | 319.2 | 1.77 | 100 | 486 | 56
45 | MP | 102 | 328.3 | 3.44 | 100 | 412 | 63
46 | MP | 26 | 612.3 | 1.77 | 100 | 551 | 70
47 | MP | 96 | 609.8 | 1.76 | 100 | 575 | 62
48 | MP | 111 | 619.3 | 4.01 | 100 | 525 | 76
49 | MP | 100 | 11.1 | 1.77 | 100 | 248 | 48
50202020$P$ fixed with $T$ increasing from 2.2 - 4.0 K as a natural warm-up. | MP | 72 | 622.3 | 3.30 | 100 | 547 | 86
51 | MP | 96 | 593.7 | 2.16 | 100 | 576 | 71
52 | MP | 94 | 589.7 | 3.13 | 100 | 565 | 69
53181818Three different voltage ramp rates: 50, 100, and 200 V/s. | MP | 974 | 10.8 | 1.76 | 50, 100, 200 | 271 | 72
54 | MP | 148 | 10.7 | 1.76 | 200 | 303 | 83
55888Holding time measurements. Refer to text for further details. | MP | 1364 | 10.6 | 1.76 | 200 | - | -
5688footnotemark: 8 | MP | 1149 | 10.5 | 1.76 | 200 | - | -
5788footnotemark: 8 | MP | 1168 | 10.5 | 1.76 | 200 | - | -
5888footnotemark: 8 | MP | 994 | 10.5 | 1.76 | 200 | - | -
61 | EP | 127 | 223.0 | 1.79 | 200 | 621 | 77
62 | EP | 93 | 11.8 | 1.79 | 200 | 368 | 86
63 | EP | 129 | 225.8 | 3.16 | 200 | 545 | 93
64191919Additional 26 holding time measurements. | EP | 97 | 422.7 | 1.79 | 200 | 814 | 63
65 | EP | 129 | 104.2 | 1.79 | 200 | 684 | 87
66 | EP | 39 | 231.1 | 1.80 | 200 | 771 | 78
67 | EP | 114 | 225.9 | 1.79 | 200 | 767 | 100
68 | EP | 134 | 229.6 | 3.17 | 200 | 574 | 128
69 | EP | 131 | 115.8 | 1.80 | 200 | 688 | 116
70 | EP | 130 | 119.1 | 2.74 | 200 | 510 | 104
Table 2:  Summary of LSHV datasets.  Set | Electrode polish | P (Torr) | T (K) |  $\mkern 1.5mu\overline{\mkern-1.5muE\mkern-1.5mu}\mkern 1.5mu$ (kV/cm)
---|---|---|---|---
1 | EP | 600.0 | 3.98 | 112.5
2 | EP | 210.0 | 3.10 | 100.0
3 | EP | 117.5 | 2.73 | 100.0
4 | EP | 29.0 | 2.07 | 75.0
5 | EP | 18.0 | 1.91 | 60.0
6161616Pressurized. | EP | 58.0 | 2.07 | 90.0
7 | EP | 600.0 | 3.98 | 100.0
8 | EP | 28.8 | 2.07 | 73.0
9 | EP | 16.2 | 1.88 | 67.0
10 | EP | 6.9 | 1.65 | 57.0

(a) EP, $<20$ Torr

(b) EP, $100-200$ Torr

(c) EP, $>200$ Torr

(d) HP, $<20$ Torr

(e) HP, $200-400$ Torr

(f) HP, $>500$ Torr

(g) MP, $<20$ Torr

(h) MP, $300-400$ Torr

(i) MP, $\sim 600$ Torr

(j) MP, $130-140$ Torr

(k) MP, $\sim 480$ Torr

(l) MP, $\sim 620$ Torr

Figure 6: (a)-(l) Distributions of breakdown field for select SSHV datasets.

####  III.2.2 Ramp rate dependence

It is natural to ask if there are time scales involved in DC electrical
breakdown and if so, what they are. For example, Ref. 25 considers a
probability density function $\mu(E)$ of breakdown initiation per unit time
interval per unit area on an electrode surface. Introducing such a quantity to
attempt to describe the statistical aspect of breakdown data is in fact quite
reasonable if the electrode surface properties are completely identical
everywhere on the electrode surface. In this case, the inception probability
becomes a function of the electric field strength only and the probability of
breakdown inception on a small surface element is independent of what happened
previously and what happens in other parts of the electrode. If this is the
case, and if $\mu(E)$ is a smooth function of $E$, then the DC breakdown field
distribution should depend on the ramp rate. (For example, if we increase the
ramp rate, we can get to higher voltage before breakdown occurs.) Motivated by
this, we measured the breakdown field distribution for different ramp rates.
We chose ramp rates of 50 V/s, 100 V/s, and 200 V/s. The results are shown in
Fig. 7.

Figure 7: Distribution of breakdown fields for three different voltage ramp
rates from SSHV dataset MP:53, showing the absence of any statistically
significant differences due to a variable ramp rate.

The data do not show a clear ramp rate dependence. Rather, the mean breakdown
voltages of the three data sets are consistent with each other within the
uncertainties of the measurements.

####  III.2.3 Distribution of time to breakdown

In order to further address the question of the time scales are involved in DC
breakdown, we performed a measurement of the time between applying an electric
field to be a certain predetermined value and the occurrence of breakdown. We
followed the following procedure for this measurement:

  1. 1.

Ramp the voltage at a constant rate towards a preset target value (first
ramp).

  2. 2.

If a breakdown occurs before reaching the target value, then record the
voltage at which the breakdown occurred and go back to step 1. The
distribution of the breakdown voltage for these events are indicated in blue
in Fig. 8.

  3. 3.

If the target voltage is reached without a breakdown, then hold the voltage
there until a breakdown occurs or until a predetermined amount of time (2 min)
elapses. If a breakdown occurs, then the time between the target voltage
reached and the occurrence of breakdown is recorded. The distribution of this
time is plotted in Fig. 9.

  4. 4.

If a breakdown does not occur in Step 3, then ramp the voltage down to zero,
and ramp up again until a breakdown occurs (2nd ramp). Record the voltage at
which breakdown occurs. The distribution of the breakdown voltage for these
events are indicated in red in Fig. 8.

  5. 5.

Repeat these steps.

We performed such a measurement for target voltages of 12, 14, and 16 kV,
corresponding to electric field strengths of 273, 318, and 364 kV/cm. These
data correspond to datasets MP:55, MP56, MP:57, and MP:58. The three target
voltages were randomly interspersed in these data sets to avoid the
possibility of a “conditioning” effect causing a trend that would depend on
the target voltage. In Fig. 8, a scaled breakdown distribution from dataset
MP:53, taken under a similar condition, is superimposed.

We make the following observations:

(1) When a breakdown occurs in the first ramp, the breakdown voltage
distribution below the target voltage agrees with that from MP:53 (scaled).

(2) Once the target voltage is reached, in a small fraction of cases a
breakdown follows with a relatively small amount of time ($\sim$20 s). In the
majority of cases, a breakdown is not observed within the observation time.

(3) In the 2nd ramp, which happens for cases where a breakdown is not observed
during the hold (step 3), a breakdown is not observed below the target voltage
except for a few cases. The breakdown distribution of the 2nd ramp agrees with
that from dataset MP:53 in the high end of the distribution. The discrepancy
between the second ramp and dataset MP:53 in the part above the target voltage
can be attributed to the events in which a breakdown is observed during the
hold (step 3).

As shown in Sec. III.2.4, the breakdown voltages among successive ramps have
little or no correlations for the majority of the datasets. And yet, the
results presented here appear to indicate that the outcome of a ramp can
depend on the outcome of the previous one, for which there is no breakdown
observed, in that the breakdown voltage in the 2nd ramp is above the voltage
up to which a breakdown was not observed in the 1st ramp. It is not entirely
clear what these observations mean to the underlying mechanism of electrical
breakdown. This is a subject of future studies along with the time scales
involved in the phenomenon.

Figure 8: Distribution of breakdown fields for events in which breakdown
occurred during the 1st ramp before reaching the preset target value (blue)
and for events in which breakdown occurred during the 2nd ramp (red). The
three panels correspond to three target voltages: (a) 12 kV, (b) 14 kV, and
(c) 16 kV. Figure 9: Distribution of time between setting the field to be a
preset value and the occurrence of breakdown. The three panels correspond to
three target voltages: (a) 12 kV, (b) 14 kV, and (c) 16 kV. A single
exponential function gives a poor fit to the data. A sum of two exponential
functions fit the distribution much better. The obtained two time constants
are shown in each plot.

####  III.2.4 Correlation and conditioning

Figure 10: Breakdown sequences for SSHV datasets MP:53 and EP:2-5.

In Fig. 10 the breakdown sequences for SSHV datasets MP:53 and EP:2-5 are
shown. These two datasets will be the focus of more detailed statistical
analysis and discussion in Sec. IV. Importantly, such an analysis is made much
more tractable when the data meet certain conditions. One such condition is
the stationarity of the data, which requires the mean, variance, and
autocorrelation to be constant over time. The time series plot in Fig. 10 show
that both these datasets have constant mean and variance. For instance, the
mean of the first 100(50) breakdowns differ only by a few percentages from the
mean of the last 100(50) breakdowns for dataset MP:53(EP:2-5), and this
difference is within the statistical uncertainty.

Correlation tests can be performed on each of dataset in Table 1. Many
statistical tests exists for such a task, including two relatively simple ones
in the Durbin-Watson test[26] and the Ljung-Box Q-test[27]. These correlation
tests do have some difficulties. In particular, both tests assess the
residuals of the breakdown field time series for correlation, and so the
results have a strong dependence on the regression model used in the analysis.
Furthermore, any temperature or pressure drifts inside the experimental volume
can complicate the analysis. The result of applying those tests shows that
about half of the datasets exhibit no correlations, a quarter are
statistically inconclusive, and the remaining quarter display some
correlations. For instance, the tests hint at possible correlations for
datasets MP:53 and MP:54 but no correlations for other datasets (e.g., MP:31,
MP:34, MP:35, MP:41, MP:49) acquired under the same conditions. The consensus
result of all the datasets does not show a definitive presence of
correlations, and the reason for the lack of complete consistency across
datasets is not clear. However, the stability of operating conditions,
particularly the pressure (see Sec. III.2.5 ), during the acquisition of each
dataset could be playing a significant role in determining the presence or
absence of correlations.

Conditioning effects on electrodes are observed in vacuum systems as well as
those containing dielectric media. The presence and magnitude of the effect
depend on both the intrinsic properties of the electrode, including its
composition and surface roughness, as well as extrinsic factors such as the
capacitance and stored energy inside the system, that is ultimately released
during a breakdown. For the datasets shown in Fig. 10, no signs of
conditioning effects are seen up to nearly 1000 breakdown sequences for
datasets MP:53 and over 350 breakdown sequences for dataset EP:2-5. Based on
the results described above, we assume stationarity of these datasets for the
analysis presented in this paper.

Finally, the measurements in Sec. III.2.3 together with the findings in this
section suggest the possibility that breakdown in LHe may be described by a
Markov chain process, in which the future breakdown is determined only by the
present state of the electrode, and not by previous breakdowns. However, a
better understanding of the effects of conditioning and how changes to the
electrode surface condition resulting from a breakdown or an applied stress
over a time period will require more dedicated measurements.

####  III.2.5 Pressure and temperature dependence

The pressure and temperature dependence of breakdown in LHe have been
previously explored by other experimenters. Meats[36] investigated breakdown
for temperatures above 4.2 K and pressures up to 7500 Torr and showed a clear
dependence of breakdown field on the applied pressure. Similarly, Chigusa _et
al._ [72] observed a significant increase in the breakdown voltage
corresponding to an increased pressure between 750 to 1500 Torr for both
uniform and non-uniform field configurations. With sphere-sphere electrodes,
Gerhold[70] measured a rise in the breakdown field strength with pressure
above one atmosphere. The same trend is observed by Burnier _et al._[74] who
also utilized sphere-sphere electrodes and performed measurements above one
atmosphere.

For measurements conducted at pressures below an atmosphere, Hara _et al._[71]
and Davidson[69] showed that the trend persists, with a reduction in breakdown
strength with decreasing pressure. A pressure dependence is also seen for
measurements made with AC power by Fallou _et al._[73] and pulsed power by
Yoshino _et al._[75]. Altogether, these observations of the pressure
dependence may be interpreted as related to gas discharge within a vapour
bubble, and we will revisit this point in Sec. IV.6.

At 1.7 K and between $\sim$10 and 600 Torr, we observed a factor of
approximately 2.5 increase in the mean breakdown field for the MP datasets as
shown in Fig. 11. These results are in close agreement with those of
Davidson[69] acquired at the same temperature and pressure range. For the SVP
data, the factor is slightly lower at approximately 2.1 and 1.9 for the SSHV
and LSHV data, respectively (Figs.11 and 12). This may be due to a small
temperature dependence but could also be indicative of the effects of boiling
liquid at 4 K which is known to reduce the breakdown strength.

(a) Temperature dependence

(b) Pressure dependence

Figure 11: The mean breakdown field of SSHV datasets as a function of (a) the
temperature and (b) the pressure. Data were acquired at both SVP and with
pressurization to allow for separation of the temperature and pressure
dependences.

(a) Temperature dependence

(b) Pressure dependence

Figure 12: The mean breakdown field of LSHV datasets as a function of (a) the
temperature (b) the pressure. All data were acquired at SVP with the exception
of the single point at 2.07 K.

####  III.2.6 Electrode surface dependence

The dependence of the breakdown field on the electrode surface finish is
evident from Fig. 11(b). Here, the mean breakdown field at a given pressure is
about a factor of two higher for the electropolished datasets as compared to
the mechanically-polished datasets. For the mechanically-polished electrodes
utilized in our measurements, the surface was polished using a lapping film
with average particle size in the range of $1-10$ $\mu$m. The electropolished
electrode surfaces should have much smaller feature sizes, however, the
surface profile was not measured.

It must be noted that attributing the difference between these two types of
electrodes solely to surface roughness is not necessarily accurate. Rather, it
is more proper to characterize the surface by its “condition” rather than just
its topographical roughness. To qualify the meaning of the term “surface
condition”, we utilize the definition proposed by May and Krauth[28], in which
surface condition is characterized by other aspects in addition to surface
roughness such as the electrode material, oxidation layer (thickness and
composition), and the presence of microcracks. The oxidation layer can depend
on a variety of factors, including the exact chemical composition of the
electrode, the chemical treatment the surface has undergone, the amount of
time and environment the electrode has been exposed for oxidation, etc.
Furthermore, various processes such as machining, stress-corrosion from
exposure to the ambient atmosphere, and local temperature gradients created
during the cooling process can generate microcracks on the electrode surface.
Together, all of these factors can complicate any comparison between the
surfaces of different electrodes.

The mechanical polishing process, beyond the likelihood of embedding abrasives
during the polishing process, can lead to changes in the material properties
such as a local increase in the mechanical strength of the surface. The
electropolishing process, even more so than mechanical polishing, can cause
changes to surface composition. In this chemical surface finishing technique,
metal is electrolytically removed and an undisturbed, metallurgical clean
surface is exposed. The metals that make up the stainless steel alloy are
removed at different rates during the process. Chromium atoms are removed at a
much lower rate than nickel and iron, resulting in a surface rich in chromium.
When exposed to oxygen, a very thin passive oxide layer is formed on the
surface, but this thin layer prevents further diffusion of oxygen into the
base material and thus prevents corrosion. These surface composition changes
can potentially lead to a corresponding change in the surface work function,
which can ultimately affect charge emission from the surface. Depending on the
nature of the change to the work function, an enhancement or suppression of
the breakdown initiating process is possible.

In this regard, it is worth noting that the work function for typical metals
tend to be in the range of $4.0-5.0$ eV [29] whereas the difference in work
function between a metal and its oxide can be as high as 2.0 eV[30]. Moreover,
just as the local microscopic field on the surface of the electrode can be
significantly different than that of the applied macroscopic field as a result
of micro-protrusions that vary in both size and shape on the surface, the work
function is not necessarily uniform and constant across the electrode surface
or with the externally applied field[31]. For fields of $10^{8}$ V/cm or
greater, work function changes above 0.5 eV are possible[32].

####  III.2.7 Scale dependence

It is well known that electrical breakdown in dielectric liquids has a
dependence on the size of the electrodes [43, 67, Gerhold1998, 60, 61, 62].
This dependence is often referred to as the “area-effect” and the “volume-
effect” in which the breakdown field decreases with increasing electrode area
and volume of the liquid under high electric fields. For uniform field
electrode geometries and constant electrode areas, the latter can be viewed as
an effect due to the inter-electrode spacing or the so-called “gap size”
effect. As such, the electrode geometry is an important consideration when
endeavoring to independently study these effects in a systematic manner. For
example, in a sphere-plane or sphere-sphere electrode geometry, both the
surface area of the electrodes and the volume of the medium subjected to high
fields change when the distance between the two electrodes changes.
Consequently, attributing the contribution of each effect to the observed
breakdown behavior can be challenging.

This issue has lead to some inconsistencies in published results regarding the
existence of a gap size effect for electrical breakdown in LHe. Several
previous studies[33, 34, 35, 36, 28, 37] have shown that the breakdown voltage
increases approximately linearly with gap size, whereas the results from
others[38, 39] have suggested a possible square-root law as found in vacuum
breakdown. At the same time, it has also been suggested that whenever
spherical electrodes are used, a hidden area effect could emerge[40, 41].
Hence, for such scaling studies, the area of the electrode subjected to high
fields must be carefully quantified, particularly in the case of spherical and
point geometries.

But even in the situation of a uniform field geometry where the volume can be
changed independently of the area, the definition of the “stressed” or
“active” area as the region in which a breakdown can occur is not necessarily
straight-forward given that breakdown is initiated over a range of electric
fields and is not solely restricted to the highest field region. Typically in
the literature, an “effective” stressed area or volume is defined as the
region where the electric field intensity is greater than 90%, or some other
percentage, of the maximum value. However, the width of a breakdown field
distribution at a given pressure can be quite broad as shown in Fig. 6. For
instance, in SSHV dataset EP:02 acquired with a electropolished electrode, the
ratio between the highest and lowest breakdown field is about a factor of
eight. This would imply that when a breakdown occurs at the maximum of the
distribution, the area of the electrode with a field higher than the minimum
of the distribution is larger by some amount than when a breakdown occurs at
lower fields. The stressed area, hence, is a function of the applied field,
and the effect would be most pronounced for geometries involving spherical or
point electrodes. Given that the probability for breakdown is field dependent,
the contribution of this additional area is not purely geometric. The shape of
the electrodes will play a critical role in determining the importance of this
effect, with the implication being that the shape of the breakdown field
distribution may be influenced by the geometry of the electrodes.

In an effort to account for this effect, some investigators[42] have defined a
statistical stressed electrode area/volume when the breakdown field
distribution is described by a Weibull[Weibull51] distribution. But even this
definition is not entirely general considering that the shape of the breakdown
distribution is not known a priori. Rather, it is more appropriate to define
the statistical stressed area of the electrode in reference to the empirically
acquired breakdown field distribution. This implies that a proper
quantification of the stressed area necessitates a sufficiently well-measured
breakdown distribution. From this consideration, the stressed area should be
viewed as a measured quantity rather than a calculated geometric value.
Therefore, our aim is to point out that caution is warranted when studying the
scaling behavior and, in particular, when comparing and interpreting data from
different experiments possessing dissimilar electrode geometries.

For near uniform-field electrodes, the stressed area may be determined by
examining the distribution of craters on the electrode surface created by the
breakdowns. An analysis of the spatial distribution of craters created by
breakdowns on the surface of a mechanically-polished electrode used in our
measurements finds that nearly all of the craters are contained within a
circular region of $\sim$5 mm radius. This region gives approximately the same
stressed area as a region with $\geq$70% of the maximum field value.
Therefore, to facilitate comparison of results from other experiments, we will
adopt this working definition of the electrode stressed area. For the SSHV HV
and ground electrodes, the stressed areas are 0.74 cm2 and 0.725 cm2,
respectively. Similarly, for the LSHV electrodes, the stressed areas are 1126
cm2 and 1125 cm2.

A comparison of the SSHV and LSHV results shows the effect of electrode area
scaling on electric breakdown in LHe. As shown in Figs.11 and 12, the mean
breakdown field strength of the LSHV data is about a factor of four lower than
that of the SSHV data at the same pressure. This observation suggests that
breakdown is a surface phenomenon that is dependent on the size of the
electrode. The lower breakdown fields measured in the LSHV apparatus, which
has an electrode $\sim$1500 times larger than that of the SSHV electrode, is
consistent with the trend observed in previous experiments on the scale
dependence of breakdown. A more detailed discussion of area scaling will be
given in Sec. IV.3.

It should be noted that the electrode spacing in the LSHV apparatus is a about
factor of seven times larger than the SSHV apparatus, and the surface
conditions of the electrodes employed in the two apparatuses are similar but
not necessarily identical. For these reasons, it is not possible to
definitively separate out the relative contributions of the area and spacing
effects or the surface quality to the observed scaling behavior. But to the
best of our knowledge, there is no conclusive evidence for a spacing effect in
uniform field electrodes, and published data indicate that the scaling
behavior is dominated by the area effect. We will discuss a possible
explanation for the apparent gap dependence observed by some experimenters in
more detail in Sec. IV.5.

##  IV Analysis and discussion

###  IV.1 Statistical approach

In the analysis of experimental data, there are often two approaches that are
adopted: a statistical approach and a physical model-based approach. Each
approach has it own merits, and a complete understanding of a problem will
necessarily require the union of both approaches. Nonetheless, when the
physical mechanism for the phenomenon under study is not well known, which is
indeed the case for electric breakdown in LHe, the statistical approach is
more suitable for use in interpretation of the data. A statistical analysis
will necessarily remove us somewhat from the underlying physical mechanism of
breakdown, which is the principal interest for experimentalists. However, it
is valuable from the standpoint of identifying patterns and attributes in the
data that can assist in constructing models to describe the phenomenon.

The statistical analysis method of choice frequently employed in studies of
electrical breakdown is a subset of the field of order statistics known as
extreme value statistics. The most commonly used function in this field is
certainly the one originally introduced by Weibull [44], which is an empirical
distribution function that has been shown to be one of the three allowable
forms of the asymptotic extreme value functions [46]. The second form of these
functions most often encountered in applications is called the Gumbel
function. These two functions have been extensively used in studies of
breakdown in dielectric media, while a third form, called the Fréchet
function, is closely related to the Weibull function but differing from it in
the range of validity. The Fréchet function, like the Weibull function, has a
finite bound whereas the Gumbel function is two-side unbounded. The choice of
which function to use depends on the phenomenon and property being studied.

However, imprudent use of those functions in the analysis of experimental data
can have considerable drawbacks. Because the parent distribution from which
the data are drawn cannot be guaranteed to be known a priori, limiting the
analysis of the data to using the three extreme value distributions can
introduce a bias in the conclusions obtained from such an analysis. It is
clear that any analysis built upon unfounded assumptions at the outset can
produce unreliable predictions. But above all, it can be quite detrimental in
the quest to further our understanding of the physics behind electrical
breakdown.

A statistical analysis of data most likely does not capture all the
information contained within it. For instance, although routinely done, it is
generally true that when empirical data are binned there is an inherent loss
of information. This is especially important for small datasets where binning
can result in amplified fluctuations and an obscuration of the true shape of
the distribution, particularly in the tail regions where the data are sparse.
The loss of information can in some instances be compensated for with
knowledge of the underlying distribution, but when the distribution is not
known, the loss can be significant. Hence, to retain the maximum amount of
information, it is preferable to analyze the data using the cumulative
distribution function (cdf) rather than as a binned histogram; the latter
being more common in the analysis of experimental data.

Absent the assumption of a model distribution, the analysis must start with
the most general formulation for breakdown statistics. Following Choulkov[45],
for an electrode with surface area, $S_{0}$, composed of small area elements,
$dS$, the total number of surface elements on the electrode is $N=S_{0}/dS$.
Let $W$ be the probability density of breakdown initiation on a small element
of the electrode, then the probability of breakdown on a single small element
is $p=WdS=WS_{0}/N$. Correspondingly, the probability that a breakdown does
not occur (i.e., the survival probability) on the surface element is $1-p$.
According to weakest link theory, the absence of a breakdown on the entire
electrode surface depends on the absence of breakdown on all of its elements;
that is, no surface element undergoes breakdown. Thus, the electrode survival
function, $P_{s}$, is the product of the probabilities of survival for each
surface element,

$\displaystyle
P_{s}=\prod_{i=1}^{N}(1-p)=\left(1-p\right)^{N}=\left(1-S_{0}W/N\right)^{N},$
(1)

where the second equality follows from the assumption that $p$ does not depend
on the surface element (i.e., the uniform case). In the limit
$N\rightarrow\infty$, Eq. 1 reduces to

$\displaystyle P_{s}=e^{-S_{0}W}.$ (2)

The complement of the survival function is the breakdown cumulative
distribution function, [44, 45]

$\displaystyle P_{b}=1-P_{s}=1-e^{-S_{0}W}.$ (3)

For the most general case in which $p$ is not uniform but depends on the
location of the surface element, the cdf can be shown to be given by

$\displaystyle P_{b}=1-P_{s}=1-e^{-\int{WdS}}$ (4)

from substitution of Eq. 2 for $p$ in Eq. 1 and taking the limit of large $N$.

The quantity of special significance in Eqs. 3 and 4 is $W$. In general, the
cdf can be dependent on a variety of factors such as the electrode surface
condition (roughness, work function, micro-particles, surface inclusions,
absorbed gas layers, microcracks, etc.), the temperature and pressure of the
liquid, and distance between electrodes. But for a given dataset, these
parameters are held fixed, so $W$ is a function of only the applied electric
field, $E$.

In survival analysis, $W$ is often referred to as the hazard function (or
failure rate) and can be interpreted as the “instantaneous rate” of a
breakdown(failure) at field $E$, but note that in the strict mathematical
sense, the hazard function is not a true probability since it can be greater
than 1. More properly, the hazard function is a conditional probability
density that the event has not yet occurred prior to field $E$ and is a non-
negative function that uniquely defines the distribution function. The
integral $\int{WdS}$ is known as the cumulative hazard function, and the
behavior of the distribution function and its scaling properties are
determined by this quantity. With the appropriate choice for $W$, the cdf can
become any one of the three extreme value distributions. For instance, the
three parameter Weibull distribution function[44] is obtained when

$\displaystyle W(E)\propto\left(\frac{E-E_{T}}{\alpha}\right)^{m},$ (5)

where $\alpha>0$, $E_{T}<E$ , and $m>0$ are known as the scale, location, and
the shape parameter, respectively. If instead, $W$ follows an exponential
function,

$\displaystyle W(E)\propto\exp{\left(\frac{E-\alpha}{\beta}\right)},$ (6)

the result is the Gumbel distribution function [46].

Therefore, the specific form of extreme value distribution appropriate for
describing the data depends upon the functional form of the continuous
(initial) distribution that the chosen property (e.g., microscopic field
intensity) would have if the sample size (electrode area) were infinite. In
particular, it depends upon the way in which the probability density decays as
the chosen extreme limit is approached. If this initial distribution density
falls off exponentially or more rapidly (e.g., a Gaussian function), then the
appropriate form is the Gumbel distribution. In the case that it falls off as
a power law, then the Weibull or Fréchet form is the most appropriate form.
However, absent a fundamental reason for breakdown data to follow one of the
three extreme value distributions, the initial density can take many other
functional forms, and the data should be the final determinant on the
appropriate form. We discuss the method for this determination below.

The cumulative hazard function can be estimated by determining the empirical
cumulative distribution function (ecdf), $\hat{P_{b}}$, from the data. In the
absence of censored data, the ecdf can be determined very simply. But for the
case in which the data is censored, several methods exists to address this
situation, and the Kaplan-Meier product limit estimate [47] and the Nelson-
Aalen estimator [48, 49, 50] are two of the more versatile methods.

The Kaplan-Meier estimator of the survival function, $\hat{P_{s}}$, is given
by

$\displaystyle\hat{P_{s}}(E)=\prod_{j:E_{j}\leq
E}\left(1-\frac{B_{j}}{N_{j}}\right),$ (7)

where $B_{j}$ is the number of breakdowns that occurred at field $E_{j}$ and
$N_{j}$ is the number of breakdowns that is yet to occur up to field $E_{j}$.
We take SSHV dataset MP:53 (mechanically polished, 1.76 K, 10.8 Torr), which
includes data for all three ramp rates and is the largest dataset taken at one
set of conditions, to illustrate the analysis approach. In Fig. 13(a), the
survival and empirical cumulative distribution functions derived from this
dataset are shown. Note that the estimator is a step function, hence, only
piecewise-continuous. However, we can define a piecewise linear function with
breakpoints at the midpoints of the jumps in the ecdf estimate. The first and
last linear segments must then be extended beyond the data to make the
function reach 0 and 1. After which the estimated cumulative hazard function
for a uniform field is determined from Eq. 3 to give

$\displaystyle S_{0}\hat{W}=-\log\left(1-\hat{P}_{b}\right).$ (8)

(a) Survival and empirical distribution functions

(b) Cumulative hazard function

Figure 13: (a) The survival and empirical cumulative distribution functions
obtained from the Kaplan-Meier product limit estimate for SSHV dataset MP:53.
(b) The estimated cumulative hazard function, $S_{0}\hat{W}$, for the same
dataset. Data from all three ramp rates are included in these results.

Figure 13(b) shows the cumulative hazard function obtained from Eq. 8. Note
that $S_{0}$ is the effective stressed area of the SSHV electrode. We have
purposely left the value of this quantity unspecified as it is not necessarily
a constant that is independent of the applied electric field. A more detailed
discussion of the surface area was given in Section III.2.7. Thus far, we have
not made any assumptions regarding the form that this function must take, but
below we will examine the possibilities that are justified from physical
considerations, starting with what the electrode surface condition can reveal
about this function.

###  IV.2 Electrode surface condition

A full characterization of the surface condition of an electrode requires the
consideration of many different factors. However, the one that is the point of
focus for most studies on electrical breakdown is the roughness of the
electrode surface. This is not surprising since roughness is a very tangible
concept and there is an intuitive notion of its effect on electrical
breakdown.

The roughness of an electrode surface may be characterized by the distribution
of surface asperities. Because each asperity give rise to a local enhancement
of the field, it is then natural to consider the distribution of this
quantity, $\beta$, which is defined by

$\displaystyle\beta=\frac{E_{\mu}}{E},$ (9)

where $E_{\mu}$ is the local (i.e., microscopic) field at the emission site
and $E$ is the macroscopic applied electric field. Larger values of the field
enhancement factor correspond to asperities on the electrode surface with
higher aspect ratios.

The hazard function, $W$, as previously discussed is a function of the
electric field. But in particular, the field that is of interest is the
microscopic field on surface of the electrode rather than the macroscopic
applied field. This interpretation is supported by the results from the
mechanically-polished electrode in comparison with those from the
electropolished one acquired under the same conditions. The difference in
breakdown field between the two electrode polishes must be a consequence of
dissimilar surface conditions. This then leads to the question of what is the
functional form for $W$.

Our qualitative understanding of the physics of breakdown in LHe can be used
to provide guidance as to the functional form of the hazard function.
Breakdown is believed to be initiated by electrons tunneling from the cathode
electrode at asperities where the field is enhanced. When the heat introduced
into the liquid by the tunneling current becomes sufficiently large for a
vapor bubble to form at an asperity, breakdown ensues. This physical picture
strongly suggests that the hazard function is most appropriately expressed as
a function of the tunneling current, $I$, and following Weibull[44], that it
is linearly proportional to the current, $W(I)\propto I$. We have no
measurement of the tunneling current at breakdown, which is estimated be less
than than a picoamp, but it can be related to the field through the Fowler-
Nordheim equation[51, 52] for tunneling of electrons from a metal to vacuum or
an insulator.

The current from Fowler-Nordheim field emission, $I$, can be expressed as[53]
rCl I & = A_e 1.54 ϕ 10^4.52 ϕ^-1/2 (βE)^2
×exp( -6.53 ×104ϕ3/2βE ) A. Here, $A_{e}$ is the effective emission area in
cm2, $\beta$ is the local field enhancement factor, $\phi$ is the electrode
work function in eV, and $E$ is the applied electric field in kV/cm.

In fitting the cumulative hazard function with Eq. IV.2, the value of the
parameter $\phi$ is known and can be fixed. We set $\phi=5.4$ eV, motivated by
two considerations: the work function for stainless steel is about 4.4 eV and
the energy of a non-localized electron in LHe is $\sim$ 1.0 eV higher than in
vacuum [55]. These factors effectively bring the work function of the surface
to 5.4 eV.

(a) Fits of the cumulative hazard function

(b) Fowler-Nordheim plot

(c) Comparison with field emission in LHe

(d) Comparison of electrode surface polish

Figure 14: (a) Multiple functional fits to the estimated cumulative hazard
function, $S_{0}\hat{W}$, for dataset MP:53. (b) The Fowler-Nordheim plot of
SSHV dataset MP:53 obtained by treating $S_{0}\hat{W}$ as a “current”. (c) The
estimated cumulative hazard function of SSHV dataset MP:53 overlayed with
measurements of field emission currents in LHe from Halpern and Gomer [57]
that have been scaled to best match the breakdown data. Refer to text for more
details. (d) Comparison of the estimated cumulative hazard functions for SSHV
mechanically-polished dataset MP:53 and the SSHV electropolished datasets
EP:(2-5).

The fit of the Fowler-Nordheim equation in Eq. IV.2 to the empirically
obtained cumulative hazard function is shown in Fig. 14(a). The goodness of
fit of the Fowler-Nordheim equation to data is much greater than either the
exponential function or the two parameter power function, and returns fit
parameters of $\beta=1444$ and $A_{e}=1.6\times 10^{-12}$ cm2. The field
enhancement factor obtained here is similar to what is found in vacuum field
emission from stainless steel electrodes of $\beta\sim 200-1000$[56]. The
interpretation of the emission area obtained from the fit is, however, not
straightforward due to the proportional relationship between the hazard
function and the current222The current obtained from Eq. IV.2 at the mean
breakdown field is the order amps whereas the actual current is below a
picoamp. The constant of proportionality must be quite small. Consequently,
any comparison to literature values is at best tenuous, but values of $\sim
10^{-16}-10^{-14}$ cm2 have been obtained in a previous experiment[56].

If the quantity, $S_{0}\hat{W}$, is treated as a “current” and plotted in the
Fowler-Nordheim plot as shown in Fig. 14(b), the data is observed to follow a
linear trend accordant with the behavior for vacuum field emission from metal
surfaces[54]. This result and the fit of the data to the Fower-Nordheim
equation provide strong support to the hypothesis that field emission is
involved in the breakdown initiating process.

Thus far, several pieces of evidence point to the connection between the field
emission current and the cumulative hazard function. This connection is more
substantial when we compare our empirical function with measurements of field
emission in LHe made by Halpern and Gomer [57]. In Fig. 14(c), the cumulative
hazard function derived from SSHV dataset MP:53 is plotted along with the
current($I_{HG}$) vs. voltage($V_{HG}$) measurements of Halpern and Gomer
[57]. Their measurements have been scaled in both variables to best match our
data, that is, $I_{HG}\rightarrow qI_{HG}$ and $V_{HG}\rightarrow kV_{HG}$,
where $q$ and $k$ are the scaling factors with a value of $5.2\times 10^{6}$
and $9.3\times 10^{-2}$, respectively. The resemblance between the shape of
their current vs. voltage measurement curve and the cumulative hazard function
further indicates a strong connection between the two.

A comparison can also be made between the cumulative hazard functions for
different electrode surface polishes as shown in Fig. 14(d). The comparison
shows that the mechanically-polished electrode can be viewed approximately as
an area-scaled electropolished electrode. This suggests a better
representation of the size parameter is $\sigma_{d}S_{0}$ with $\sigma_{d}$ as
the density of breakdown initiation sites. So the relevant size parameter to
consider is the number of initiating sites rather than the surface area, and
such an interpretation is reasonable from the point of view of surface
roughness. A consequence of this size effect is seen in the difference of the
breakdown distribution widths shown in Fig. 6, but a more detailed discussion
of this is given in Sec. IV.3.

Although Fig. 14(d) shows considerable similarity in the shape of the
cumulative hazard function between the mechanically-polished and
electropolished datasets acquired under similar conditions, there are notable
differences. These are apparent from analyzing the fit to dataset EP:2-5 which
is shown in Fig. 15. It is found that the best fit is obtained by using a sum
of two Fowler-Nordheim functions. The field enhancement factor of
$\beta_{2}=1571$ obtained for one of the functions is close to that obtained
from dataset MP:53, but the effective emission area, $A_{e,2}$, is about a
factor of ten lower, which is consistent with the comparison shown in Fig.
14(d). For the data points above 450 kV/cm, the addition of a second function
with enhancement factor of $\beta_{2}=431$ provides the best fit.

The result of this exercise of characterizing the breakdown in LHe as being
the result of asperities with two distinct enhancement factors is not intended
to provide an accurate representation of the physical reality of the electrode
surface. Rather it demonstrates that breakdown is not necessarily limited to
sites having the largest field emission current. It illustrates the inability
of the statistical approach in determining the microscopic nature of the
electrode surface. Further, it suggests the possible limitations of extreme
value theory and of the description of the emission current through the
Fowler-Nordheim equation. For instance, at sufficiently high fields and high
current densities, deviations from the standard Fowler-Nordheim
description[54] and surface work function changes[32] are known to exist.

Figure 15: A fit of electropolished electrode dataset EP:2-5 to a two term
Fowler-Nordheim equation.

Based on the results presented, we will adopt the interpretation of the
proportionality between the Fowler-Nordheim field emission function and the
hazard function as a working hypothesis in the discussion that follows. Our
present understanding of the breakdown mechanism in LHe, although still
qualitative in nature, provides strong motivation for such an interpretation,
and which we showed is supported by our data. What’s more, the hypothesis has
broad enough scope with the potential for establishing a unified model to
explain breakdown in different dielectric materials and other related studies.
Thus, it serves as a practical starting point to guide exploration in a
direction that may lead to a better understanding of the processes behind
these phenomena, for up until the present time, the connection between the
statistics of breakdown to the underlying physical mechanism has not been
clearly established.

###  IV.3 Area scaling

It is common in the literature to explore the scaling behavior of the
breakdown field with electrode surface area by fitting measurements of
experiments having different electrode areas. However, dissimilarities in
setups, conditions and methodologies may obscure underlying scaling laws and
other features of a comparison. It is preferable to compare data taken in
different experiments whose conditions are similar and well characterized. But
absent the availability of such information, one can also explore the scaling
predicted by a single well-characterized experiment.

In the most general case where no assumptions are made on the form of the
cumulative hazard function, it is possible to determine the area scaling
behavior by utilizing the framework outlined in Sec. IV.1. Let $n$ be defined
as a scaling factor so that the area-scaled empirical cumulative distribution
is expressed as

$\displaystyle\hat{P_{b}}(E,n)=1-e^{-nS_{0}\hat{W}},$ (10)

where $nS_{0}\hat{W}$ is determined from a dataset with an arbitrary choice
for $n$. Here, we will take $n$ to be one so that scaling is performed
relative to the surface area of the SSHV electrode. The $n=1$ dataset is what
we will refer to as the reference dataset, or reference electrode.

Figure 16: The electrode surface area scaling of the empirical cumulative
distribution function for different values of the scaling factor, $n$. The
$n=1$ reference curve is determined from dataset MP:53 in the SSHV data using
the procedure described in Sec. IV.1. The dotted curves are obtained by
fitting the reference dataset and applying the scaling factors.

The area-scaled empirical cumulative distribution function for several
different values of the scaling factors is shown in Fig. 16. The $n=1$ data
corresponds to the reference surface area of the SSHV electrode while the
other values represent electrodes with surface areas that are different than
the SSHV electrode by a factor $n$. From Fig. 16, the median breakdown field
($\hat{P}_{b}=0.5$), or any other percentile, can be immediately read off from
the curves to determine how it scales with the electrode surface area. There,
however, are clearly statistical limits to this data-based approach as
exemplified by the $n=10^{-1}$ and $n=10^{3}$ curves. The extent of the
limitation is primarily determined by the statistics in the reference dataset.
As a result of this, when the value of $n$ is much larger or smaller than
$n=1$, the median, or other chosen percentile, breakdown field is no longer
well-constrained by the data points. However, this would also imply that with
sufficient statistics in the reference dataset, the data-based scaling can be
carried out to an arbitrarily sized electrode. Thus, the method discussed here
represents a robust, model-independent approach to area scaling, avoiding
possibly invalid assumptions regarding the underlying breakdown distribution.

(a) Mechanically-polished electrode

(b) Electropolished electrode

Figure 17: The probability density function, $p(E)$, derived from fitting the
cumulative hazard function for (a) SSHV dataset MP:53 (mechanically-polished,
10.8 Torr, 1.76 K) and (b) the combined SSHV dataset EP:2-5 (electropolished,
12.8 Torr, 1.81 K) for different scaling factors, $n$. The experimentally
acquired breakdown field distribution for the datasets are also plotted. The
statistics for the curves are given in Table 3.

(a) Mechanically-polished electrode

(b) Electropolished electrode

Figure 18: Electrode area scaling of the mean and $p^{\textrm{th}}$ percentile
breakdown fields determined from fitting (a) the 10.8 Torr (1.76 K),
mechanically-polished electrode SSHV dataset MP:53 and (b) the 12.8 Torr (1.81
K), combined electropolished electrode SSHV datasets EP:2-5 using a Fowler-
Nordheim function for the cumulative hazard function. The mean breakdown
fields of the SSHV data used in the scaling are shown along with the mean
breakdown fields of the LSHV data of 57 kV/cm and 67 kV/cm at 6.9 Torr and
16.2 Torr, respectively. The data point at 105 kV/cm from the MSHV apparatus
(arrow) represents the highest field achieved using electropolished
electrodes, but at which no breakdown was observed.

There are several important features in Fig. 16 to highlight. First, as the
area of the electrode is scaled up, the width of the breakdown field
distribution becomes narrower. A comparison of the SSHV and LSHV data shows
agreement with this finding. Conversely, scaling down to smaller areas results
in a broader distribution. This implies that more breakdown statistics are
needed to characterize the breakdown field at a given confidence level when
the electrode used in the measurement is small.

Figure 16 also shows that the effect of scaling on the tails of the
distribution function is dependent on the direction of the scaling. When the
area is scaled up(down) the data points are pushed towards the high(low) field
region on the curve. This suggests that scaling up may present somewhat of a
obstacle for reliability analysis, which is typically more concerned with the
breakdown field at low percentiles as this allows for safe operational
boundaries and parameters of a device to be established in order to reduce its
failure risk. But when the breakdown field distribution is nearly symmetric
about the median field, it is possible to use the high percentile breakdown
field values (e.g., 90%, 95%, 99%), that are still well-constrained by the
data to estimate the low percentile (e.g., 1%, 5%, 10%) breakdown fields. If
the assumption of symmetry cannot be applied to the distribution, the skewness
must also be taken into account in a more careful determination.

The change in the distribution width and the median breakdown field with
electrode area may be understood without reference to any particular breakdown
mechanism in the following way: An electrode can be viewed as a union of
multiple sub-electrodes that we can categorize as one with low-field breakdown
initiation, medium-field breakdown initiation, and so on. Importantly, the
division and categorization of these sub-electrodes are made in the
statistical, or probabilistic, sense and is in reference to its breakdown
initiating probability rather than a geometric one. These sub-electrodes can
also be viewed as sub-groups of electrode surface states. An overall scaling
on the electrode scales each of these sub-groups in the same manner. For
larger electrode sizes, the sub-group leading to low-field breakdown
initiation is similarly larger than for a smaller electrode. As the electric
field is ramped up, the contribution to the total probability of breakdown due
to the low-field sub-group becomes high enough that a breakdown occurs before
reaching the higher fields needed to initiate breakdown in the other sub-
groups. In a more mathematical illustration, the hazard function, $W$, can be
expressed as a linear combination of many terms, arranged in order from low-
field to high-field breakdown initiation. These terms are related to the
distribution of surface protuberances and their associated field enhancements.
When the area, $S_{0}$, is scaled up, the leading term dominates and
determines the outcome of the particular voltage ramp sequence. Thus, when the
electrode is large, the properties of the observed breakdown distribution is
determined primarily by the low-field sub-group (i.e., the “weakest sub-
group”), and this explains the lower breakdown fields and narrower
distributions encountered in larger-sized electrodes.

The above picture implies a type of self-censoring333Here, we say the data is
censored because there is only partial information about the distribution
function, that is, the value of the function above some boundary field is
unknown. of the data when going in the reverse direction, that is, scaling
from a large-sized reference electrode down to a smaller one. This is because
the predominance of the low-field breakdown initiating sub-group prevents
experimental probing of the higher-field sub-groups. Therefore, no information
on these sub-groups is obtained by the experiment, leading to the appearance
of a censoring of the empirical distribution function. This would explain the
very peculiar behavior of the $n=10^{-1}$ curve in Fig. 16 where the
distribution function terminates abruptly, signaling an end to the region of
accessible information. As a consequence, area scaling is a non-symmetric
operation in the sense that the direction matters just as does the scaling
factor. The implication for experimental studies of electrical breakdown is
that more complete information on the breakdown distribution function, or more
precisely, the cumulative hazard function, may be obtained by employing
smaller-sized electrodes rather larger ones, which on the surface may
certainly appear quite counter-intuitive. But such an approach will also have
trade-offs that must be considered, and these will be more evident in the
discussion that follows.

Table 3:  Statistics for the area-scaled probability density functions for datasets MP:53 and EP:2-5. $n$ is the scaling factor, $\mu$ is the mean breakdown field, FWHM is the full-width at half maximum, and $\gamma$ is the skewness. Set | $n$ |  $\mu$ (kV/cm) | FWHM (kV/cm) | $\gamma$
---|---|---|---|---
MP | $10^{-1}$ | 522.8 | 446.7 | 0.441
MP | $10^{0}$ | 269.6 | 177.8 | 0.240
MP | $10^{1}$ | 164.1 | 82.5 | 0.033
MP | $10^{2}$ | 112.5 | 43.9 | -0.142
MP | $10^{3}$ | 83.7 | 26.0 | -0.281
MP | $10^{4}$ | 65.8 | 16.8 | -0.389
EP | $10^{-1}$ | 839.2 | 666.6 | 0.213
EP | $10^{0}$ | 438.6 | 327.5 | 0.085
EP | $10^{1}$ | 244.3 | 159.5 | 0.127
EP | $10^{2}$ | 150.2 | 75.4 | 0.023
EP | $10^{3}$ | 103.2 | 40.2 | -0.144
EP | $10^{4}$ | 76.8 | 23.9 | -0.282

The statistical limits of the empirical-based scaling method are apparent.
However, by a suitable choice of fitting function for the cumulative hazard
function in Eq. 10, these limits may be remedied. We showed previously that
the estimated cumulative hazard function is well-fitted by a function of the
form given by the Fowler-Nordheim field emission equation. In Figs.17(a) and
17(b), the probability density function, $p(E)$, derived from fitting the
associated dataset in each figure is shown. The empirical breakdown field
distribution is shown along with the fitted function for $n=1$, and the
probability density functions for other values of $n$ are also plotted. The
statistics for the scaled functions are tabulated in Table 3. The skewness is
defined as $\gamma=\mu_{3}/\mu_{2}^{3/2}$, where $\mu_{3}$ and $\mu_{2}$ are
the third and second central moments, respectively.

With increasing $n$, the mean breakdown field decreases and a narrowing of
$p(E)$ is observed. These features are consistent with those displayed by the
data-based scaling of the cumulative distribution function shown in Fig. 16.
But also note the change in the shape of the function and, in particular, its
skewness with scaling factor. The function shifts from positive to negative
skewness as $n$ increases. This is a rather intriguing feature that is not
immediately evident from Fig. 16, but is simply a consequence of the shape of
the cumulative hazard function. This unique characteristic can provide for a
strong test of the validity of our area scaling analysis beyond that of the
how the breakdown field scales with electrode area, which often have
uncertainties associated with surface conditions and definition of stressed
area.

In Fig. 18, the predicted breakdown field as a function of the stressed area
derived from analysis of the mechanically-polished dataset MP:53 and the
combined electropolished datasets EP:2-5 is shown. The breakdown data acquired
in the LSHV apparatus and the limit obtained from our Medium Scale High
Voltage (MSHV) system [7] are also plotted in the figure for comparison. The
mean breakdown field measured in the LSHV apparatus is about 22% and 36%
smaller than the predicted values obtained from the mechanically-polished and
electropolished electrode dataset, respectively. This represents very good
agreement between measurements and predictions considering the large scaling
factor of about $10^{3}$ and uncertainties associated with defining the
stressed area, the limited statistics in the LSHV dataset, and differences in
surface conditions. Concerning the latter point, note there exists large
variations of over 30% in the mean breakdown field between SSHV mechanically-
polished datasets acquired under essentially identical conditions (e.g.,
$\sim$1.7 K and 10 Torr) as shown in Fig. 11.

Finally, it is important for us to remark on the fundamental limitations of
statistical area scaling. It is evident that the scaling of the electrode is
realized only in the mathematical sense, in which we conceptualize the
replication, i.e., making _identical_ copies, of the reference electrode $n$
times to produce the surface area of the scaled electrode. There exists an
inherent sampling bias in this operation, so that the statistically scaled
electrode is not necessarily a perfect representation of the physical
electrode. For instance, one can conceive of a type of defect that causes a
breakdown to occur at low fields. If this class of defect has a sufficiently
low density, it is more likely to be absent in small electrodes as compared to
larger ones, implying that the selection bias will tend to give slightly
higher predicted breakdown fields for very large electrode areas than what
will be observed experimentally. In the end, the above considerations show
that there is no true substitute for experimental measurement.

###  IV.4 Area scaling in other noble liquids

In studies of the area effect in liquid argon[60, 61, 62] (LAr) and xenon[62]
(LXe), the breakdown field, $E$, as function of stressed area is often modeled
by the expression, $E=CA^{p}$, where $C$ is a constant, $A$ is the stressed
area and $p<0$ is the power index. This scaling relation can be obtained by
assuming that the hazard function follows a two-parameter power-law, or
equivalently that the cumulative breakdown distribution function is given by
the Weibull function. For LAr, $p$ values between $-0.22$[61, 62] and
$-0.26$[60] are found from a global fit of experimental data with stressed
areas in the range of approximately $10^{-4}-10^{1}$ cm2. With LXe, a scarcity
of breakdown data exists, but a $p$ value of $-0.13$[62] is found by fitting
the results of two experiments[65, 62, 63]. However, the reliability of this
fit value is quite questionable given the uncertain experimental conditions
for the data point from Ref.65. In addition, the data in Ref.62, 63 show non-
stationarity, indicating a possible conditioning effect, which makes the
interpretation of the data less straightforward.

The breakdown field area-scaling expression, $E=CA^{p}$, is to an extent
problematic. Clearly, in the limiting behavior, the breakdown fields obtained
from such an expression are physically unreasonable, that is, it becomes
arbitrarily small with increasing electrode surface area. It is possible,
though, to maintain the validity of the expression by restricting the domain
of applicability, but such an accommodation brings about other questions. For
instance, what is the scaling behavior outside the range of validity and why
is it only valid in the given range? Is the true scaling law different that
what is being assumed?

Figure 19: LAr breakdown field versus stressed area, defined as the area with
an electric field strength greater than 90% of the maximum field strength. The
area scaling curves are determined by applying the analysis method proposed in
this work to the breakdown data from Acciarri _et al._[60] acquired with a
stressed area of $\sim 2$ cm2. Data from past LAr experiments and the often
assumed scaling curves, $E=CA^{p}$, are also plotted for comparsion.

To address those questions and further explore the applicability of the
proposed analysis method in this work to experimental datasets in other noble
liquids, we utilize the LAr breakdown data from Acciarri _et al._[60]. Worth
noting is that this particular dataset contains only around 170 breakdowns and
is also binned. Therefore, its statistical power is significantly lower than
that of our LHe datasets. The data from Acciarri _et al._ were acquired with a
sphere-plane electrode geometry where the sphere was 76 mm in diameter and the
90% stressed area is $\sim 2$ cm2. The curves representing the mean,
10-percentile, and 90-percentile breakdown field as determined from applying
our analysis method to this single dataset is shown in Fig. 19. Also plotted
are the measurements from past studies of LAr breakdown as well the often
assumed scaling expression, $E=CA^{p}$, where the values of the parameters are
from Auger _et al._ [61] and Tvrznikova _et al._ [62].

The 76-mm Acciarri _et al._ data[60] (green triangles), taken for various
stressed areas, are well fitted by our scaling curves, determined from data
for one single stressed area, as shown in Fig. 19. These curves also have
better agreement with the data of Tvrznikova _et al._ [62] as compared to the
$E=CA^{p}$ scaling curves, which overpredict the breakdown fields for the
76-mm data but underpredict for the Tvrznikova _et al._ data. The Auger _et
al._ [61] data also show good agreement with our scaling curves However, the
agreement of these curves with the smaller stressed area data is lower, but a
likely explanation for this is the difference in surface condition of the
electrodes used to acquire those data. Specifically, the 5-mm and 1.3-mm
Acciarri _et al._[60] data were acquired with electrodes with a chrome finish
whereas the 76-mm electrode had a mirror-finish. A comparison of the 5-mm and
76-mm data at around a stressed area of 0.1 cm2 indicates such an effect,
which also appears to be present in the Blatter _et al._ [64] data acquired
with different electrode sizes. This highlights the point previously made
about the problematic nature of fitting data from a group of experiments with
variable and sometimes uncertain experimental conditions.

###  IV.5 Electrode gap spacing dependence

Breakdown in dielectric liquids is shown to be a surface phenomenon that does
not involve the bulk properties of the liquid, yet many experiments have been
interpreted as showing a electrode gap spacing dependence. Such a dependence
is often expressed in the form $V=kd^{s}$, where $V$ is the breakdown voltage,
$k$ is a constant, $d$ is the gap spacing, and the index $s$ characterizes the
strength of the dependence. However, there is no generally accepted mechanism
of breakdown that would form the basis for such a relationship, and some
experimenters[39, 67] have suggested an alternative relation of
$V=V_{0}+ha^{1-j}$. But inconsistencies in experimental results have thus far
prevented any satisfactory resolution to this problem.

The possibility that natural sources of radioactivity can give rise to a gap
spacing dependence is inconsistent with experimental observation that show
electrons, $\alpha$ particles, and the daughter nuclei from radioactive
sources such as 241Am electroplated directly onto high voltage electrodes do
not initiate breakdown for applied fields up to 45 kV/cm in LHe[58, 59]. By
considering the density of energy deposition from the daughter nucleus of an
241Am decay, a lower bound on the minimum energy density needed for the
initiation process can be determined. The lack of breakdown initiation from
these highly ionizing particles show they cannot form a basis to explain any
presumed spacing dependence.

Experiments performed by Galand[34] and Lehmann[35] found that the breakdown
voltage in LHe is approximately proportional to the electrode gap spacing up
to 2.5 mm. Results from Mathes[33] using 12.7-mm diameter steel-sphere
electrodes are similar for gap spacings up to 1.25 mm. However, significant
gap dependence for spacings between $10-100$ $\mu$m are indicated by the
measurements of Goldschvartz and Blaisse [66] using a 10-mm diameter sphere-
plane (tungsten-stainless steel) electrodes.

Measurements by Schwenterly _et al._[39] utilized stainless steel sphere-plane
electrodes with a diameter of 38 mm and gap spacings between $0.25-2.5$ mm.
Their results showed that breakdown fields increase for gap spacings below
about 1 mm but become nearly constant for spacings above 1.5 mm. The results
of Gerhold[68] made with spherical electrodes of 50-mm diameter and gaps up to
0.5 mm are consistent with those of Schwenterly _et al._[39] where for gap
spacings above 1.5 mm, the breakdown fields are nearly constant.

Experiments employing larger-sized spherical electrodes or uniform-field
electrodes have observed much milder gap spacing effects. For instance,
Meats[36], using uniform-field electrodes of 60-mm diameter observed that for
spacings less than 1.0 mm, observed a slight dependence of breakdown field on
$d$. But at larger spacings, the breakdown voltage varied linearly with a gap
spacing up to nearly 4 mm. Measurements made by Meyerhoff[37] with a polished
250-mm diameter brass sphere and brass ground plane for spacings up to 12 mm
found that the breakdown voltage increases nearly linearly with spacing up to
200 kV. In contrast, the results of Fallou _et al._[38] indicate a pronounced
departure from linearity with $s=0.5$ up to $d=10$ mm, implying a “square-root
law” as found in vacuum breakdown. These measurements, however, were made with
small-diameter spheres and had spacings greater than the sphere diameter.

Table 4: Data from previous experiments on the gap spacing dependence of breakdown in LHe by Goldschvartz and Blaisse[66], Schwenterly _et al._[39], and Meyerhoff[37]. All data were acquired at 4.2 K and SVP and utilized a sphere-plane electrode geometry. Both the boiling(non-boiling) results from Meyerhoff are shown. Gap (mm) | Stressed area (mm2) |  $\overline{E}$ (kV/cm)
---|---|---
0.010777Goldschvartz and Blaisse | 0.137 | 3760
0.02077footnotemark: 7 | 0.266 | 3100
0.03077footnotemark: 7 | 0.396 | 2480
0.04077footnotemark: 7 | 0.536 | 2280
0.05077footnotemark: 7 | 0.688 | 2010
0.06077footnotemark: 7 | 0.826 | 1980
0.07077footnotemark: 7 | 0.955 | 1850
0.08077footnotemark: 7 | 1.077 | 1700
0.09077footnotemark: 7 | 1.203 | 1650
0.10077footnotemark: 7 | 1.378 | 1500
0.25191919Schwenterly _et al._ | 12.59 | 800
0.501919footnotemark: 19 | 25.99 | 655
0.751919footnotemark: 19 | 38.42 | 569
1.001919footnotemark: 19 | 50.82 | 530
1.251919footnotemark: 19 | 66.65 | 500
1.501919footnotemark: 19 | 79.49 | 480
1.751919footnotemark: 19 | 93.35 | 445
2.001919footnotemark: 19 | 108.39 | 425
2.251919footnotemark: 19 | 118.98 | 420
2.501919footnotemark: 19 | 135.77 | 415
3.0131313Meyerhoff | 987.63 | 173.8(238.1)
4.01313footnotemark: 13 | 1392.40 | 176.0(241.1)
5.01313footnotemark: 13 | 1761.40 | 163.5(224.0)
6.01313footnotemark: 13 | 2196.90 | 151.5(207.5)
7.01313footnotemark: 13 | 4156.10 | 161.1(220.7)
8.01313footnotemark: 13 | 2670.90 | 161.5(221.2)
9.01313footnotemark: 13 | 2783.20 | 159.9(219.1)
10.01313footnotemark: 13 | 3102.00 | 158.7(217.4)
11.01313footnotemark: 13 | 3226.80 | 155.8(213.4)
12.01313footnotemark: 13 | 4156.10 | 152.3(208.7)

Figure 20: The mean breakdown field as a function of the stressed area as
measured by Goldschvartz and Blaisse [66], Schwenterly _et al._[39], and
Meyerhoff[37]. For the Meyerhoff measurements, the breakdown fields in non-
boiling LHe are about 37% higher than the the fields in boiling liquid. Also
shown are the scaling curves determined from analysis of the SSHV data and the
measured breakdown fields at 4.0 K in the LSHV apparatus.

Although the inconsistencies in experimental results are quite evident, some
patterns do emerge. In particular, the magnitude of the observed effect
appears to be associated with the geometry of the electrodes (sphere-sphere vs
sphere-plane and uniform-field) as well as the size of the electrodes relative
to the gap spacing. As we previously discussed in Sec.III.2.7, these elements
have a direct and considerable impact on the characterization of the electrode
stressed area. Hence, it is conceivable that the spacing effect is merely a
consequence of an unaccounted for area effect.

To investigate that possibility, we re-analyzed the data from Goldschvartz and
Blaisse [66], Schwenterly _et al._[39], and Meyerhoff [37]. The geometries of
these experiments (sphere-plane) are modeled in the COMSOL Multiphysics
software package [24] to determine the stressed area, defined to be the region
on the electrode surface with a field strength greater than 70% of the maximum
field, for each of the gap spacings measured in their experiments. The choice
of this stressed area definition is motivated by the discussion in Sec.
III.2.7 concerning the SSHV electrode stressed area. The results of these
calculations are shown in Table 4 along with the experimental measurements. In
Fig. 20, the measured breakdown fields from these three experiments as well as
those from the LSHV apparatus are plotted as a function of the calculated
stressed area. Also shown are the predicted scaling curves at 4.0 K as
determined from analysis of the SSHV measurements for the mechanically-
polished and electropolished datasets at $1.7-1.8$ K. The curves at the lower
temperatures are extrapolated to 4.0 K by applying a factor that is determined
from the data in Fig. 11. Both the electropolished and mechanical-polished
curves are plotted due to the fact the polish of electrodes used in some of
the previous experiments were not clearly specified.

Remarkably, the measurements of Goldschvartz and Blaisse [66], Schwenterly _et
al._[39], and Meyerhoff [37] closely follow the predicted area scaling
behavior. It is particularly important to re-emphasize that these measurements
cover a wide range of gap spacings, 10 $\mu$m to 12 mm, that essentially
encompasses the full range of gap sizes in all previous experiments exploring
this dependence. Thus, we have a reasonable expectation that the results of
other experiments not included in our re-analysis may be explained in similar
fashion. Also of note is that measurements made with non-boiling LHe by
Meyerhoff are stated as being 37% higher than those made in boiling liquid. In
Fig. 20, the non-boiling measurements are determined by applying a 1.37 factor
to the boiling measurements included in the Meyerhoff paper. Related to this
observation is that although the low breakdown fields from the LSHV
measurements in relation to the scaling curves may be explained by a surface
quality effect, it may also be suggestive of possible boiling effects present
when those measurements were made. Altogether, the re-analysis of the gap
spacing measurements from these previous experiments that assumed to
demonstrate a gap dependence can be satisfactorily shown to be the consequence
of an areal dependence. This is in keeping with liquid dielectric breakdown
being a surface phenomenon. The finding also lends further support to the
validity of our area scaling analysis.

###  IV.6 Temperature and pressure dependence

A qualitative understanding of the temperature and pressure dependence of
electrical breakdown in LHe may be realized by considering the following
simple picture. If electrical breakdown is initiated by the formation of
bubbles, then the stability of the bubble against collapse is a necessary
condition for a breakdown to develop. The bubble development process is
connected to the heterogeneous nucleation process in LHe; the latter may also
be tied to the stochastic nature of the breakdown phenomenon. Nevertheless,
further consideration of this topic will be deferred to a future study. When
the bubble is created by the injection of energy into the liquid resulting
from field emission on the electrode surface, the amount of energy emitted
must be sufficient to counteract the opposing forces of surface tension and
externally applied pressure that work to prevent the bubble from expanding and
ultimately leading to breakdown. However, only a fraction of the total emitted
energy, $W_{e}$, can go into opposing those two forces because the energy is
partitioned into many components to drive different energy processes [76] such
as ionization, vaporization of the medium, and light production.

Some of the processes are dependent on both the temperature and pressure of
the liquid, while others are independent of both quantities. If we consider
the special case of constant temperature, all the pressure independent terms
can be combined into the term, $W_{0,T_{0}}$, and the pressure dependent term
can be expressed as $W_{P}(P)$, giving

$\displaystyle W_{e}=W_{0,T_{0}}+W_{P}(P).$ (11)

This simplification is motivated by our observation that the breakdown field
is primarily dependent on the pressure of the liquid rather than its
temperature. The temperature dependence appears to be a sub-dominant effect,
but is certainly not negligible as we discuss below.

The energy injected into the liquid is provided by the field emitted electrons
which have a distribution of energies. The current density is obtained by
integration of this energy distribution[77, 78]. Therefore, the total-energy
of emission should be proportional to the emission current.

The total input energy, however, depends on the timescale of the emission
process. Experiments that use impulse power (ns to $\mu$s) instead of DC
generally observe higher breakdown fields, indicating that the timescale is an
important factor to consider. But without explicit knowledge of this quantity
we make a simplifying assumption that this timescale is constant over the
pressure and temperature range under consideration here. As a consequence, the
total energy resulting from field emission is taken to be proportional to the
emission current.

With that, Eq. 11 can be expressed as

$\displaystyle W_{0,T_{0}}+W_{P}(P)\propto
E^{2}\exp\left(\frac{-D\phi^{3/2}}{\beta E}\right).$ (12)

Defining the parameter in the exponential as $B=D\phi^{3/2}/\beta$, we obtain

$\displaystyle W_{0,T_{0}}+\frac{4\pi R^{3}P}{3}\propto
E^{2}\exp\left(\frac{-B}{E}\right),$ (13)

where the work due by the external pressure, $P$, on a bubble of radius, $R$,
is $4\pi R^{3}P/3$ and $E$ is the applied field expressed in units of kV/cm.
Finally, the externally applied pressure is related to the breakdown field by

$\displaystyle P=AE^{2}\exp\left(\frac{-B}{E}\right)-C_{T_{0}},$ (14)

with $A$, $B$, and $C_{T_{0}}$ as positive constants.

Calculating the numerical values of the constants $A$ and $C_{T_{0}}$ from
first principles is very difficult because of the simultaneous, complicated
dynamical processes occurring during bubble development, but the constant $B$
is related to the work function and field enhancement factor. If we use the
numerical values of these parameters obtained from fitting $S_{0}\hat{W}$ for
SSHV dataset MP:53, we obtain $B=567$ kV/cm. However, we will not fix $B$ to
this value because the field enhancement factor varies from dataset to dataset
and should viewed only as an average effective value. Notably, at high fields,
the exponential factor in Eq. 13 approaches one and the breakdown field scales
as the square root of the applied pressure. We note that a similar pressure
dependence for breakdown in LHe is suggested by Meats [36].

Figure 21: Black-line: Fit of breakdown data acquired at $T\approx 1.7$ K with
Eq. 13; Red-squares: SSHV 1.7 K data points used in the fit; Open-blue-
circles: SSHV 4 K SVP data; Orange-triangles: Pressurized data from Meats;
Dotted and dash-dotted lines: Empirical pressure dependence relationship from
Meats for two surface tensions. The Meats data points and empirical curves are
re-scaled using Fig. 18(a) to account for the difference in electrode surface
areas between experiments.

Shown in Fig. 21 are the mean breakdown fields for mechanically-polished SSHV
datasets that were acquired at a temperature of $\approx 1.7$ K over a range
of pressures. In addition, the breakdown fields for datasets acquired at
$\approx 4.0$ K and SVP are also plotted. An interesting observation is that
at the same pressure the typical breakdown field is lower at 4.0 K than it is
at 1.7 K. This does not appear to be merely the result of statistical scatter
because similar behavior is observed at other temperatures and pressures (see
Figs. 6(h), 6(j), 6(k), and 6(l)) where the lower temperature data at a given
pressure has the higher average breakdown field. This difference could be due
to the effects of boiling liquid on measurements acquired at SVP. But the
trend in Fig. 21 is also consistent with the change in the surface tension and
latent heat of vaporization between the two temperatures plotted (Table 5).
Both of these quantities decrease between 1.7 K to 4.0 K, and this results in
a reduction in the constant $C_{T_{0}}$ in Eq. 13. Physically, this suggests
that less energy is required to vaporize the liquid and counteract the surface
tension of the bubble for the same externally applied pressure. Thus, for a
given breakdown field, the corresponding pressure increases, which effectively
causes the data points in Fig. 21 to shift to the right.

Table 5: Selected properties of liquid helium at different temperatures[79]: saturated vapor pressure, $P$, surface tension, $\sigma_{s}$, and latent heat of vaporization, $L$. T (K) | P (Torr) |  $\sigma_{s}$ (N/m) |  $L$ (J/mol)
---|---|---|---
5.0 | 1470.1 | $1.570\times 10^{-5}$ | 47.67
4.6 | 1064.3 | $5.160\times 10^{-5}$ | 72.59
4.2 | 744.3 | $8.954\times 10^{-5}$ | 83.19
4.0 | 612.2 | $1.095\times 10^{-4}$ | 87.00
3.5 | 352.9 | $1.626\times 10^{-4}$ | 92.84
3.0 | 180.4 | $2.161\times 10^{-4}$ | 94.11
2.5 | 76.7 | $2.623\times 10^{-4}$ | 92.50
2.17 | 37.2 | $2.862\times 10^{-4}$ | 90.75
1.7 | 8.5 | $3.224\times 10^{-4}$ | 91.91
1.2 | 0.6 | $3.426\times 10^{-4}$ | 84.14
0.7 | $<0.1$ | $3.508\times 10^{-4}$ | 74.35
0.4 | $<0.1$ | $3.536\times 10^{-4}$ | 68.17

In addition to the SSHV data, the pressurized data from Meats [36] are also
plotted in Fig. 21. The Meats data were acquired with 6-cm diameter uniform-
field electrodes ($S\approx 20$ cm2). Their electrodes were prepared with
metal polish, so the surface condition should be most comparable to the
mechanically-polished SSHV electrode. To account for the electrode surface
area difference between the experiments, a scaling factor obtained from Fig.
18(a) is used to re-scale the Meats data.

The dotted and dashed-dotted curves in Fig. 21 are calculated from the model
for pressure dependence proposed by Meats [36] in which the breakdown field in
V/m is given by:

$\displaystyle
E=\frac{0.20}{\sqrt{\epsilon_{0}}}\left(P+\frac{2\sigma_{s}}{5\times
10^{-9}},\right)^{1/2},$ (15)

where $P$ is the externally applied pressure in Pa and $\sigma_{s}$ is the
surface tension in N/m. The curve for the Meats model with the surface tension
at 5 K provides an adequate fit to both datasets. However, the 1.7 K surface
tension curve has a large discrepancy with our SSHV data.

In Fig. 21, the solid black curve represents the fit of the 1.7 K SSHV data to
Eq. 14 and has good agreement with the SSHV data as well as the data from
Meats, especially considering the uncertainties in electrode conditions
between experiments. The value of the parameter $B$ in the fit corresponds to
$\beta\approx 1739$, which is close to the value of $\beta$ obtained from fits
of the cumulative hazard functions for datasets MP:53 and EP:2-5. The physical
reasonableness of the other two fit parameters is difficult to gauge without
independent estimates of their expected values. Nevertheless, the pressure
dependence of the breakdown field is shown to be well described by the simple
qualitative picture presented here.

###  IV.7 Determining breakdown for arbitrary electrode geometry

Area scaling in Eq. 10 can be extended to the general case in which the
electrode has an arbitrary field distribution. For a given potential value,
the surface of the electrodes has a distribution of field strengths. If the
area that has a field value $E_{i}$ is given by $S(E_{i})$, then its
contribution to the survival function is given by
$P_{s}=e^{-S(E_{i})W(E_{i})}$. The survival function for the entire electrode
surface is then given by

$P_{s}=\prod_{\\{E_{i}\\}}e^{-S(E_{i})W(E_{i})}.$ (16)

Here, it is understood that $S(E_{i})$ is a function of the electrode
potential and that a limit in which $E_{i}$ takes continuous values is
assumed.

We illustrate how this is done in practice with the simple example electrode
geometry shown in Fig. 22. The shape of the two identical electrodes is made
of a flat circular disk with a torus attached to the edge. The different
colors on the surface indicates the field strength at that location. As seen
the part of the torus facing the other electrode has high fields, and the flat
part in the middle has a large area with a modest field strength.

Figure 23 shows the distribution of the surface field for one of electrodes.
Three distinct peaks are observed. The small peak at $\sim$130 kV/cm
corresponds to the high field area on the surface of the torus facing the
other electrode. The peak at $\sim$80 kV/cm corresponds to the flat area in
the middle facing the other electrode. The peak at $\sim$15 kV/cm is due to
the back of the electrode, facing the grounded surface (not shown in Fig. 22).
For such a geometry it is important to take into account the contribution from
the entire electrode surface to breakdown.

Figure 24 shows the calculated probability of no breakdown (i.e., the survival
probability) given by Eq. 16 for this geometry. The value of $W(E_{i})$ has
been determined by a fit to our SSHV dataset MP:53 for mechanically-polished
(MP) electrodes and dataset EP:2-5 for electropolished (EP) electrodes.

For experiments that require large electrodes with a complex shape, an
optimization of the electrode shape can be performed to minimize Eq. 16 while
meeting other experimental requirements. This inclusive optimization method is
different that what is typically done, which is to reduce the so-called local
“hot-spots,” or small regions with high field intensities. The latter approach
overlooks the scale dependence of electrical breakdown and the larger-area but
lower field regions’ contributions to the overall breakdown probability.

Figure 22: An example electrode geometry to illustrate the utility of Eq. 16.
The shape of the two identical electrodes is made of a flat circular disk with
a torus attached to the edge. The surface field strengths shown are calculated
for a potential difference of 1 MV. Inset: the cross section of the
electrodes.

Figure 23: The field distribution calculated for the positive electrode shown
in Fig. 22. The potential difference between the two electrodes is 1 MV.

Figure 24: The calculated probability of no breakdown for the electrode
geometry shown in Fig. 22 as a function of the potential difference for the
case of electropolished (EP) and mechanically-polished (MP) electrodes.

##  V Conclusion and prospects

The electrical breakdown phenomenon in liquid helium has been investigated
using several electrodes of different surface quality. The probability
distribution of breakdown with respect to field has been analyzed using
extreme value theory. This analysis confirms that breakdown is a surface
phenomenon directly correlated with Fowler-Nordheim field emission. The
statistical theory immediately provides a means of predicting the breakdown
behavior of electrodes of different size and shape. The results of several
earlier experiments on breakdown in liquid helium are found to be consistent
with the current analysis. The implications of the present results to
measurements with other noble liquids and to arbitrary shaped electrodes are
discussed.

The connection between breakdown and field emission raises the possibility of
characterizing the breakdown properties of an electrode in a non-destructive
manner such as by relating field emission in vacuum to the breakdown
properties in noble liquids. A better understanding of the time-dependent
phenomenon, specifically the time scales involved in the breakdown process and
the effects of a holding field, is required. We leave exploration of these
topics to future studies.

##  VI Acknowledgments

This work was supported by the United States Department of Energy, Office of
Science, Office of Nuclear Physics under Contract Numbers DE-AC52-06NA25396
and 89233218CNA000001 under proposal LANLEEDM (LANL), DE-FG02-ER41042 (NCSU),
and DE-AC05-00OR22725 (ORNL) through Los Alamos National Laboratory Integrated
Contract Order No. 4000129433 with Oak Ridge National Laboratory. In addition,
this work was supported by the National Science Foundation under Grant Nos.
1506451 (Valparaiso) and PHY-1307426 (NCSU). We gratefully acknowledge the
support provided by the Physics and AOT Divisions of Los Alamos National
Laboratory. We are also grateful to Vince Cianciolo for his encouragement and
support.

## References

  * [1] J. Gerhold, Cryogenics 38, 1063 (1998).
  * [2] A. A. Belevtsev, High Temperature 53, 770 (2015).
  * [3] B. Rebel et al., JINST 9, T08004 (2014).
  * [4] R. Golub and S. K. Lamoreaux, Phys. Rep. 237, 1 (1994).
  * [5] M. G. D. van der Grinten et al., Nucl. Instrum. Methods Phys. Res. Sect. A 611, 129 (2009).
  * [6] T. M. Ito et al., Phys. Rev. A 85, 042718 (2012).
  * [7] T. M. Ito et al., Rev. Sci. Instrum. 87, 045113 (2016).
  * [8] M. W. Ahmed et al., JINST 14, P11017 (2019).
  * [9] W. Guo and D. N. McKinsey, Phys. Rev. D 87, 115001 (2013).
  * [10] T. M. Ito and G. M. Seidel, Phys. Rev. C 88, 025805 (2013).
  * [11] S. Knapen, T. Lin, and K. Zurek, Phys. Rev. D 95, 056019 (2017).
  * [12] S. A. Hertel, et al., arXiv:1810.06283v1.
  * [13] S. M. Clayton, et al., JINST 13, P05017 (2018).
  * [14] B. S. Blaisse, A. van den Boogaart, F. Erné, Bull. Inst. Intern. Froid Anexie, 1, 333 (1959).
  * [15] C. Blank and M. H. Edwards, Phys. Rev. 119, 50 (1960).
  * [16] A. A. Belevtsev, Nucl. Instrum. Methods Phys. Res., Sect. A 327, 18 (1993).
  * [17] R. Kattan, A. Denat, and O. Lesaint, J. Appl. Phys. 66, 4062 (1989).
  * [18] E. F. Kelley and R. E. Hebner, Jr., J. Appl. Phys. 52, 191 (1981).
  * [19] E. O. Forster, J. Phys. D: Appl. Phys. 23, 1506 (1990).
  * [20] The SNS nEDM experiment (V. Cianciolo and B. W. Filippone, spokespersons) [https://www.phy.ornl.gov/nedm/].
  * [21] T. M. Ito, J. Phys. Conf. Ser. 69, 012037 (2007).
  * [22] J. C. Long et al., arXiv:physics/0603231
  * [23] Janis Research, https://www.janis.com
  * [24] COMSOL Multiphysics, http://www.comsol.com.
  * [25] A. L. Kupershtokh, E. I. Palchikov, D. I. Karpov, I. Vitellas, D. P. Agoris, and V. P. Charalambakos, J. Phys. D:Appl. Phys. 35, 3106 (2002).
  * [26] J. Durbin and G. S. Watson, Biometrika 37, 409 (1950).
  * [27] G. Ljung and G. E. P. Box, Biometrika 66, 66 (1978).
  * [28] D. May and H. Krauth, “Influence of the electrode surface conditions on the breakdown of liquid helium”, IEEE Trans. MAG. Vol. 17, pp. 2089-2092, 1981; D. May, PhD. thesis, TU Karlsruhe, 1981.
  * [29] W. M. Haynes, CRC Handbook of Chemistry and Physics, 95th ed. (CRC Press, Boca Raton, Fla., 2014).
  * [30] A. K. Vijh and K. Dimoff, J. Mater. Sci. 11, 150 (1976).
  * [31] H. Qiu, R. P. Joshi, A. Neuber, and J. Dickens, Semicond. Sci. Technol. 30, 105038 (2015).
  * [32] T. T. Tsong and E. W. Müller, Phys. Rev. 181, 530 (1969).
  * [33] K. N. Mathes, IEEE Trans.EI-2, 24 (1967).
  * [34] J. Galand, Compt. Rend. 266, 1302 (1968).
  * [35] J. P. Lehmann, “Measures dielectriques dans les ftuides cryogeniquesjusqu’a 200 kV-SO Hz,” Revue Gen. Elec., 79: IS (1970).
  * [36] R. J. Meats, “Pressurized-helium Breakdown at very Low Temperatures”, Proc. of IEE, vol. 119, no. 6, pp. 760-766, 1972.
  * [37] R. W. Meyerhoff, Development of a Rigid AC Superconducting Power Transmission Line, In: Timmerhaus K.D. (eds) Advances in Cryogenic Engineering. Advances in Cryogenic Engineering, vol 19. Springer, Boston, MA (1995).
  * [38] B. Fallou et al., in: Low Temperature and Electric Power, Intern. Inst. Refrigeration, London (March 1969), p. 201.
  * [39] S. W. Schwenterly et al., “Dielectric strength of liquid helium in millimeter gaps,” Conference on Electrical Insulation & Dielectric Phenomena - Annual Report 1974, Downingtown, PA, USA, 1974, pp. 585-593.
  * [40] A. H. Sharbaugh, E. B. Cox, R. W. Crowe, and P. L. Auer, “The effect of electrode configuration on the electric strength of hexane,” 1955 Conference on Electrical Insulation, pp. 16-20, doi: 10.1109/EIC.1955.7533320.
  * [41] T. J. Gallagher, Simple dielectric liquids, (Clarendon Press, Oxford, 1975).
  * [42] H. Goshima et al., “Weibull statistical analysis of area and volume effects on the breakdown strength in liquid nitrogen”, IEEE Trans. on Diel. and Electr. Insul., vol. 2, no. 3, pp. 385-393 (1995).
  * [43] K. H. Weber and H. S. Endicott, Trans. Amer. Inst. Elect. Eng. 75, 371 (1956).
  * [44] W. Weibull, J. Appl. Mech.18, 293 (1951).
  * [45] V. V. Choulkov, IEEE T. Dielect. El. In Vol. 12 p.98 (2005).
  * [46] E. I. Gumbel, Statistics of Extreme, Columbia University Press, New York, USA, 1958.
  * [47] E. L. Kaplan and P. Meier, “Nonparametric estimation from incomplete observations”, Journal of the American Statistical Association, 53:457-481 (1958).
  * [48] W. Nelson, “Hazard Plotting for Incomplete Failure Data”, Journal of Quality Technology, 1:1, 27-52 (1969).
  * [49] W. Nelson, “Theory and Applications of Hazard Plotting for Censored Failure Data”, Technometrics, 14:4, 945-966 (1972).
  * [50] O. Aalen, “Nonparametric Inference for a Family of Counting Processes”, Ann. Statist. 6, no. 4, 701-726 (1978).
  * [51] R. H. Fowler and L. Nordheim, Proc. Roy. Soc. A 119, 173-81, 1928.
  * [52] R. H. Good and E. W. Müller, Field Emission, in “Handbuch der Physik,” Springer Verlag, Berlin, 21, 176-231, 1956.
  * [53] J. W. Wang and G. A. Loew, “Field emission and RF breakdown in high gradient room temperature linac structures,” SLAC-PUB-7684.
  * [54] G. Fursey, _Field Emission in Vacuum Microelectronics_ , (Plenum Publishers, New York, 2005).
  * [55] J. R. Broomall, W. D. Johnson, and D. G. Onn, Phys. Rev. B 14, 2819 (1976).
  * [56] M. BastaniNejad, M. A. Mohamed, A. A. Elmustafa, P. Adderley, J. Clark, S. Covert, J. Hansknecht, C. Hernandez-Garcia, M. Poelker, R. Mammei, K. Surles-Law and P. Williams, Phys. Rev. ST Accel. Beams 15, 083502 (2012).
  * [57] B. Halpern and R. Gomer, J. Chem. Phys. 51, 1031 (1969).
  * [58] T. M. Ito, S. M. Clayton, J. Ramsey, M. Karcz, C. Y. Liu, J. C. Long, T. G. Reddy, and G. M. Seidel, Phys. Rev. A 85, 042718 (2012).
  * [59] N. S. Phan, V. Cianciolo, S. M. Clayton, S. A. Currie, R. Dipert, T. M. Ito, S. W. T. MacDonald, C. M. O’Shaughnessy, J. C. Ramsey and G. M. Seidel, et al., Phys. Rev. C 102 3, 035503 (2020).
  * [60] R. Acciarri et al., JINST 9 P11001 (2014).
  * [61] M. Auger et al., JINST 11 P03017 (2016).
  * [62] L. Tvrznikova et al., JINST 14 P12018 (2019).
  * [63] L. Tvrznikova, _Sub-GeV dark matter searches and electric field studies for the LUX and LZ experiments_ , Ph.D. thesis, Yale University (2019).
  * [64] A. Blatter et al., JINST 9 P04006 (2014).
  * [65] B. Rebel _et al._ , JINST 9 T08004 (2014).
  * [66] J. M. Goldschvartz and B. S. Blaisse, Br. J. Appl. Phys. 17 1083 (1966).
  * [67] J. Gerhold, M. Hubmann, and E. Telser, Cryogenics 34 579 (1994).
  * [68] J. Gerhold, Cryogenics 12 370 (1972).
  * [69] A. J. Davidson, _HighVoltage Breakdown in the Ramsey Cell of the CryoEDM experiment - An experimental study of some relevant parameters_ , Ph.D. Thesis, University of Sussex (2011).
  * [70] J. Gerhold, IEEE Transactions on Electrical Insulation 24 2 155-166 (1989).
  * [71] M. Hara _et al._ , Proceedings 8th International symposium on High Voltage Engineering, 2 559-562 (1993).
  * [72] S. Chigusa _et al._ ,IEEE Transactions on Dielectrics and Electrical Insulation, 6 3 385-392 (1999).
  * [73] B. Fallou _et al._ , Cryogenics, 10 2 142-146 (1970).
  * [74] P. H. Burnier _et al._ , Advances in Cryogenic engineering, 15 76-84 (1970).
  * [75] K. Yoshino, K. Ohseko, M. Shiraishi, M. Terauchi, and Y. Inuishi, J. Electrostatics 12, (1982).
  * [76] R. Kattan, A. Denat, N. Bonifaci, “Formation of vapor bubbles in non-polar liquids initiated by current pulses”, IEEE Transactions on Electrical Insulation, Vol. 26, No. 4, 1991, pp. 656-662.
  * [77] R. D. Young, Phys. Rev. 113, 110 (1959).
  * [78] R. D. Young and E. W. Müller, Phys. Rev. 113, 115 (1959).
  * [79] R. J. Donnelly and C. F. Barenghi, J. Phys. Chem. Ref. Data 27, 1217 (1998).