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e3e-mail<EMAIL_ADDRESS>e1e-mail<EMAIL_ADDRESS>e2e-mail<EMAIL_ADDRESS>e5e-mail<EMAIL_ADDRESS>e6e-mail<EMAIL_ADDRESS>e7 _Present Address:_ IRFU, CEA,
Université Paris-Saclay, F-91191 Gif-sur-Yvette, France

11institutetext: Max-Planck-Institut für Kernphysik, Saupfercheckweg 1, 69117
Heidelberg, Germany

# Characterization of alpha and beta interactions in liquid xenon

Florian Jörge3,addr1  Dominick Cichone1,addr1  Guillaume Eurine2,e7,addr1
Luisa Hötzschaddr1  Teresa Marrodán Undagoitiae5,addr1  Natascha Ruppe6,addr1

(Received: date / Accepted: date)

###### Abstract

Liquid xenon based detectors have achieved great sensitivities in rare event
searches. Precise knowledge of the scintillation and ionization responses of
the medium is essential to correctly model different interaction types in the
detector including both signal and background-like ones. The response of
liquid xenon to low energy electrons and to alpha particles has been studied
in the Heidelberg Xenon (HeXe) dual-phase xenon TPC. We determine the light
and charge signal yields for keV-energy electrons and MeV-energy alpha
particles as well as the electron drift velocity for electric drift fields
between 7.5 and 1 640 V/cm. A three dimensional simulation using COMSOL
Multiphysics® is used to characterize the applied drift field and its
homogeneity.

††journal: Eur. Phys. J. C

##  1 Introduction

Over the past decades, xenon time projection chambers (TPCs) have become one
of the leading detector technologies for rare event searches, such as the
search for neutrinoless double-$\upbeta$ decay Albert:2017owj  or for the
elastic scattering of dark matter off target nuclei Undagoitia:2015gya . Key
advantages of this technology are the low energy threshold of
$\mathcal{O}(\textrm{keV})$, the ultra low background and the large masses
that can be realized. The next generation of detectors is being commissioned
or planned with masses at the multi-ton scale: XENONnT Aprile:2020vtw , LZ
LZ:2019sgr , nEXO Albert:2017hjq  or DARWIN Aalbers:2016jon .

In dual-phase liquid xenon TPCs, particle interactions within the active
detector volume can be detected via signals originating from excitation and
ionization of the medium. Prompt scintillation light (S1) is emitted in the
de-excitation of xenon excimer states created when the medium is excited by a
particle’s energy deposition. Alongside the scintillation, xenon atoms can
also become ionized. Free electrons produced at the interaction site can be
collected if an electric drift field is applied. The electrons drift
vertically and are extracted to a gas phase on top of the liquid. In the gas
phase a strong electric field amplifies the signal (S2) via proportional
scintillation Lansiart:1976 . From the time difference between the S1 and S2
signals, the depth of the interaction can be determined. For detectors with
several photosensors on top and bottom of the target, the photon hit
distribution on the sensors can be used to reconstruct the $(x,y)$-coordinates
of each event. Combined with the depth measurement, this provides a full
3-dimensional reconstruction of the event’s position.

The Heidelberg Xenon (HeXe) TPC has been developed as an R&D instrument
towards improving the understanding of various aspects relevant for current
experiments searching for rare events. The system has been previously employed
to test the response of R11410 PMTs of XENON1T to xenon scintillation light
Barrow:2016doe , to test radon reduction via boil-off distillation
Bruenner:2016ziq , and to test the impact of PTFE cleaning recipes for radon
daughter removal on TPC performance Bruenner:2020arp . In this manuscript, we
detail the TPC setup and report on the response of liquid xenon to alpha (MeV
energies from 222Rn, 218Po & 214Po) and beta (keV energies from
${}^{\mathrm{83m}}\mathrm{Kr}$) interactions as a function of electric field.
Note that only little literature is available for interactions of alpha
particles Aprile:1991xb ; Aprile:2006kx . We investigate the light and charge
yield dependence of alpha-induced energy depositions for fields between 7.5
and 1 640 V/cm. A detailed 3D electric-field model based on the finite element
method (FEM) is used to precisely determine the electric field, its
homogeneity and uncertainty.

In section 2, we describe the HeXe TPC setup and in section 3 the electric
field simulations performed in order to obtain a uniform electric field.
Afterwards, section 4 describes the auxiliary systems and section 5 the data
flow from data acquisition to data analysis. Sections 6 and 7 contain the
results including the measurements of charge yields, light yields for beta and
alpha particles, as well as a drift velocity determination for various
electric fields.

##  2 The HeXe detector

The HeXe TPC has a cylindrical drift volume of 5 cm height and 5.6 cm diameter
containing a total mass of 345 g liquid xenon (see figure 1).

Figure 1: Schematic figure of the HeXe TPC with its components.

It is instrumented with 2 Hamamatsu R6041-406 2-inch metal channel dynode
photomultiplier tubes (PMTs) specially selected for a high quantum efficiency
(QE of 39.1% and 38.8% for top and bottom, respectively). The PMTs are
typically operated between $(900-980)$ V with a negatively biased high voltage
(HV) being provided by an iseg power supply. A custom-made voltage divider is
used to distribute the HV to the PMT cathode and dynodes. The single
photoelectron (SPE) response of the PMTs can be measured in situ using
external LED boards, illuminating the two PMTs with photons via two optical
fibers. Due to the large diameter of the cryostat hosting the TPC, a PTFE
support structure has been designed in which active volume is embedded. In
this way, the amount of LXe required to fill the active volume to the desired
height is minimized. The active volume is surrounded by an easily exchangeable
PTFE cylinder, which can be used to study the impact of PTFE chemical cleaning
on the purity of liquid xenon, as has been reported in Bruenner:2020arp .

The electric fields in the TPC are generated using three hexagonal pitched
grids as described in section 3. The drift field is homogenized using a field
cage assembly consisting of three field shaping rings. Cathode and gate
electrodes are connected to them by a resistor chain using four
$1\,\mathrm{G\Omega}$ high voltage resistors with a tolerance of 1%. A custom
made CF40 flange was designed as HV feedthrough. It is equipped with three
KINGS SHV 10 kV welded HV connectors insulated with an additional PTFE piece.

In order to measure the large signals originating from alpha energy
depositions, the emitted light needs to be attenuated before illuminating the
PMTs to avoid both DAQ and PMT-base saturation. The attenuation is achieved by
using thin PTFE sheets of 700 $\upmu$m and 550 $\upmu$m thickness in front of
the top and bottom PMT, respectively. The transparency of these PTFE sheets to
liquid xenon scintillation light was determined separately and is reported in
Cichon:2020ytl .

##  3 Electric field configuration

The electric fields in the TPC are maintained using three electrode meshes.
The drift field which guides the electrons upwards, out of the interaction
site, is defined by the cathode and gate electrodes. While the drift field has
been varied in the measurements shown here, the field applied to extract
electrons from the liquid to the gas phase and amplify the charge signal
Lansiart:1976  is kept at an approximate value of $10$ kV/cm in the gas phase.
High voltage is applied to the electrodes using two iseg modules, where one of
them provides a high-precision current read out allowing for monitoring of the
cathode-to-gate current.

The electrode meshes have a hexagonal pattern and were etched out of 100
$\mathrm{\upmu m}$ thick stainless steel sheets by Great Lakes Engineering.
Each of the electrodes is housed inside a 2 mm thick circular stainless steel
frame having an outer diameter of 10 cm and an inner diameter of 5.6 cm. The
gate and cathode electrodes are rotated horizontally by $\pm 45^{\circ}$ with
respect to the anode electrode, due to construction reasons. Like the
electrode frames, the three field shaping rings are made from 2 mm thick
stainless steel with an outer diameter of 10 cm and an inner diameter of 8 cm
and are placed equidistantly along the drift length. Voltage is applied to
each electrode separately using three PTFE insulated cables guided along the
outer region of the PTFE support structure.

###  3.1 Electric field simulation

The design of the HeXe TPC has been optimized for a high electric-field
homogeneity based on a detailed detector simulation. Using the commercial
finite element software COMSOL Multiphysics® comsol:1998 , a model containing
all necessary electrical components was created. The number of field shaping
rings or the design of the electrodes (section 3.2) were optimized based on
the outcome of this simulation.

For a correct representation of the hexagonal pitched electrodes as well as
their relative rotation, a full three-dimensional geometry is chosen instead
of the more common 2D approximation. Figure 2 shows the 3D geometry as
implemented in COMSOL. The blue color scale displays the value of the
potential at each metallic surface. The colored plane represents the intensity
of the field in the target region. The complete field map, also outside the
TPC, is shown in figure 5.

Figure 2: Illustration of the COMSOL model used for electric field simulation.
Shown is the outcome for an average field of 1.25 kV/cm. While the blue color
scale represents the potential at the metallic surfaces, the orange scale
shows the field intensity across a plane through the central region. Several
selected electric field lines are illustrated in green.

In order to calculate the field in the TPC for any given combination of
electrode and PMT voltages, the superposition principle can be used. First,
the field distribution stemming from each individual electrical component is
simulated with a potential value of one volt. These results are then exported
from COMSOL on a grid of 100 x 100 x 100 points, evenly spaced in
$(z,r^{2},\varphi)$ throughout the active volume. Afterwards, the overall
field can be computed efficiently by summation of the individual fields,
scaled to the desired voltage of each respective component. This avoids the
computationally expensive re-computation of the full solution for each voltage
combination, allowing for an iterative optimization of the field homogeneity
with respect to the voltages applied to the electrodes. Residual field
inhomogeneities are quantified using the central 68% inter-percentile range of
the field magnitude from all points of the grid.

###  3.2 Electrode design

The electrodes should provide a high homogeneity of the electric field while
having the maximum possible optical transparency in order to maximize the
collection of photons. A study with an electric field simulation allows to
evaluate how the field homogeneity is influenced by different design choices
of the electrodes. The influence on the optical transparency is estimated
using the geometrical coverage.

Electrodes made from parallel wires showed in the simulation a slightly higher
field uniformity when compared to an hexagonal shaped electrode having the
same geometrical coverage. However, given its higher mechanical stability, the
hexagonal design has been favored for the HeXe setup. For these electrodes,
several combinations of the hexagonal cell spacing (pitch, p) and wire
thickness (d) were compared within ranges of
$\mathrm{p}=\left[1\,\mathrm{mm},\,...,10\,\mathrm{mm}\right]$ and
$\mathrm{d}=\left[0.1\,\mathrm{mm},\,...,1\,\mathrm{mm}\right]$ respectively.

Figure 3 compares the simulated magnitude of the electric field as a function
of the $z$-coordinate for two choices of the electrode design. The lines
indicate the median of the absolute value of the electric field, whereas the
shaded regions show the range of 68% field deviation within each slice along
the height of the active volume.

Figure 3: Electric field as function of depth ($z$-coordinate) in the TPC for
different simulated values of wire thickness and pitch, in comparison to ideal
plate electrodes (black line).

The field that would be exerted by electrodes made from solid plates is shown
in black. Since in this case there would be no field leakage through the
electrodes, it can be regarded as the limiting case of best achievable field
homogeneity with the setup. The residual inhomogeneities of about 2.1% are
then due to stray fields which are not compensated by the field-shaping rings.
Compared to this ideal case, hexagonal electrodes with a pitch of 1 mm (blue
line) show only a small amount of additional field inhomogeneities of 2.2%
while each electrode allows for a light transmission of 79.7% at the same
time. For electrodes having a pitch of 6 mm (red line), this transparency
increases to 96.7%, however the field also becomes significantly less uniform
(29.5%). This holds true for homogeneity along the z-direction as well as in
the x-y direction. The transparencies of the hexagonal electrodes are
estimated as the ratio between the uncovered surface and the total surface of
the grids. Note that by this approximation any dependence on the direction of
the incoming light is not considered.

Figure 4 shows the field deviation from the median field value as a function
of the applied field strength.

Figure 4: Dependence of the electric field uncertainty (68% relative
deviation) as a function of the applied drift field for two electrode designs.
As expected, the simulated field for smaller pitched electrodes (blue line)
shows a significantly smaller variation. Regions close to the electrodes are
removed by a similar fiducial volume selection as used for the analysis.

It can be seen that the relative uncertainty quickly increases for very small
drift fields. This is due to the unavoidable field leakage from the extraction
region in which a strong electric field needs to be maintained. At a median
drift field of 460 V/cm the relative deviation reaches a minimum of 1.2%. At
higher applied fields, the deviation slowly increases. We assume this effect
comes from the field leakage from the region below the cathode. The deviation
reaches a value of 1.5% for a drift field of 1.26 kV/cm.

Two grid options were installed in our TPC: in a first configuration the
meshes had a wire thickness of $100\,\upmu$m and a pitch of 6 mm and in the
second configuration $120\,\upmu$m and 1 mm, respectively. Figure 5 shows the
outcome of the simulation for the two configurations (6 mm pitch on the left
and 1 mm on the right) together with the corresponding data from the TPC at a
median field of approx. 400 V/cm. While the middle panels show the field in
and around the target, the upper panels are zoomed into the electron
extraction region showing the two electrodes and the quartz PMT window. The
color scale represents the magnitude of the electric field.

Figure 5: Simulation results and data for hexagonal grids of 6 mm pitch (left)
and 1 mm pitch (right). The voltages of the individual components are -900 V,
+3750 V, -5 V, -1950 V and -950 V for the top PMT, anode, gate, cathode and
bottom PMT respectively, yielding a median electric drift-field of about 400
V/cm. Top: Zoom into the extraction region of the HeXe TPC. Shown is the
magnitude of the electric field on a logarithmic scale. Steel and PTFE
components are colored in dark and light gray, respectively. Center: Magnitude
of the electric field inside the drift volume as well as within the PTFE
components of the TPC. Steel components are illustrated in gray. Bottom:
Comparison of the S2 width and S2 area resolution in data acquired with grids
of 6 mm and 1 mm pitch.

The lower panels show for each case data of the S2 pulse width (left) and the
S2 pulse size (right) as a function of the drift length in the TPC. In the
liquid-xenon target region, the second configuration (right side in figure 5)
gives a significantly more homogeneous electric field. The deviation from the
median field is $(1-2)\%$ for the 1 mm pitch and $\sim 13$% for the 6 mm pitch
(see figure 4).

The increased field homogeneity in the case of the electrode configuration
with the smaller pitch results in an enhanced quality of the data. The
variation in the S2 pulse width is strongly reduced with a standard deviation
changing from 0.40 to 0.23 in the drift region $(10-12)\,\upmu$s for the 6 mm
and 1 mm pitch, respectively. Similarly, the S2 size distribution is narrower
(see section 5 for more details on the data analysis). Note that the S2-width
increases with the drift distance due to diffusion and the S2-size decreases
as a result of electron attachment to impurities in the liquid xenon. The
correction for the latter effect is described in section 5. For all
measurements shown in this paper, we use grids with d = 120 $\upmu$m and p = 1
mm for all three electrodes.

##  4 Auxiliary systems

###  4.1 Cryogenic, purification and calibration systems

The detector is operated inside a 20.1 cm diameter vacuum insulated cryostat.
Continuous cooling is provided by a Leybold COOLPAK 6000-1 helium-driven pulse
tube refrigerator (PTR). A copper cold head attached to it is used to condense
the xenon. The temperature of the cold head is adjusted to
$T_{\mathrm{cold\leavevmode\nobreak\ head}}=-107.5\,\mathrm{{}^{\circ}C}$
using a PID controlled heater cartridge. This cryogenic system was already
operated successfully in previous measurements Barrow:2016doe ;
Bruenner:2016ziq .

A gas handling system is employed to store the xenon, to introduce calibration
sources into the detector, and to remove electronegative impurities from the
xenon. The last is necessary to maintain sufficiently large light and charge
yields as impurities like oxygen and water decrease the signal yields. The
purification is obtained by recirculating the xenon gas with a KNF double-
diaphragm pump through a hot SAES gas purifier SAES  which continuously
removes electronegative impurities from the xenon with a very high efficiency.
A schematic representation of the gas system is shown in figure 6.

Figure 6: Simplified scheme of the HeXe gas system used for xenon purification
and introduction of gaseous calibration sources.

In the purification loop, the liquid xenon is extracted from a reservoir below
the bottom PMT. Simultaneously, clean xenon is introduced to the gas phase
where it liquefies at the cold head. The liquid drops are subsequently guided
back to the liquid phase through a funnel and enter the detector through a
channel located above the bottom PMT window. The gas recirculation flow is
continuously monitored by a MKS mass flow meter and controlled to a value
about 2.75 SLPM using a valve bypassing the recirculation pump.

Opening the cryostat and performing modifications on the TPC that are relevant
for the measurements (e.g. installation of light attenuators) are carried out
while minimizing the exposure to air Bruenner:2020arp . These operations are
performed in a permanently installed glove bag under a GN2 atmosphere at a
slight over-pressure. This procedure reduces the contamination from gas
components present in ambient air that could diffuse into e.g. the PTFE of the
TPC and reduce the xenon purity.

Two calibration sources are employed for the measurements of sections 6 and 7:
a ${}^{\mathrm{83m}}\mathrm{Kr}$ electron-emitting source and a 222Rn alpha
source. Both are introduced into the TPC through the gas system as shown in
figure 6. ${}^{\mathrm{83m}}\mathrm{Kr}$ is emanated from
${}^{\mathrm{83}}\mathrm{Rb}$ loaded zeolite beads lebeda_rb83  and produces
internal conversion electrons with energies of 32.1 keV and 9.4 keV. The rate
of krypton inside the target varies between a few kHz to $\sim 100$ Hz due to
the short half life of the 83Rb source producing
${}^{\mathrm{83m}}\mathrm{Kr}$. The alpha source originates from a liquid
226Ra standard. The emanated 222Rn is extracted from the source using a flow
of helium and stored in an activated carbon trap. The trap is subsequently
attached to the gas system and can be opened to the xenon circulation flow via
the opening of a bypass. Due to the 222Rn half-life of almost 3.8 days, it
takes at least 3 weeks for the detector to reach its background level after
the introduction of the source. Typically, a rate about $100$ Hz is introduced
into the liquid xenon target.

###  4.2 Slow control system and detector stability

The gas pressure inside the cryostat, in the gas system and in the cryostat’s
insulation vacuum are measured using digital manometers and vacuum sensors.
The temperature inside the TPC is recorded with four Pt100 thermometers which
provide as well a rough estimate of the liquid xenon height during filling. A
more precise measurement of the liquid level is made by four custom-made level
meters which are distributed across the circumference of the support
structure. Consisting of a concentric assembly of an inner $1$ mm thick metal
rod surrounded by an outer cylinder of $4$ mm diameter, they allow to
determine the LXe height by relative capacitance increase. Three of the level
meters have a length of 1 cm and are sensitive to the liquid xenon levels
between the gate and the anode region. The fourth level meter, having a length
of 10 cm can be used to monitor the complete height of the drift volume.

All sensors are read out by a LabVIEW Labview  program, with the values stored
into a periodically backed-up PostgreSQL Postgresql  database for later
access. They are running on the same machine to ensure database input in case
of network outages. For each sensor, two separate sets of thresholds can be
specified (warning and alarm). If the readings pass them, emails or SMS
messages are being sent to HeXe operators to alert them. Both the program and
the database are, in turn, monitored using Nagios Nagios  which is set up on a
separate server, in order to also send alerts in case at least one of them is
not responsive or accessible. The modules providing HV for the PMTs and grids
are controlled by a tool written in Python. It continuously reads out voltage
and current values, writing them into the above-mentioned database.
Furthermore, it is capable of notifying operators of HV trips via email/SMS
and attempts to automatically ramp up tripped channels if desired. In
addition, it monitors the cryostat pressure via the database. If the pressure
drops below a customizable threshold, all HV channels are ramped down to avoid
damage due to discharges. Schedules to ramp up or down HV can also be set
making automatic voltage scans possible.

Figure 7 shows the pressure and temperature evolution inside the cryostat
during the $\mathrm{{}^{222}{Rn}}$ measurement described in sections 6 and 7.
A typical measurement starts by filling the cryostat with about 2 bar
(absolute pressure) of GXe. Subsequently, the gas is cooled down (1) in two
steps to approx. $-107.5^{\circ}$C (at the cold head).

Figure 7: Evolution of temperature (in blue) and pressure (in red) during the
$\mathrm{{}^{222}{Rn}}$ measurement. After filling the cryostat, the cool-down
phase is started (1). Additional GXe is filled and liquefied (2). After
reaching the required LXe level, the xenon recirculation is started and the
measurement begins. The measurement ends with the recuperation of the LXe (3).
About 1 bar of GXe is left in the cryostat and recirculated, to speed up the
warming-up time (4).

Next, GXe is filled and rapidly liquefied in the pre-cooled system (2). To
reach the required xenon level inside the TPC a total mass between 2.5 and 3
kg is necessary. In this state, the gas recirculation starts and the
measurement period begins. During the stable operation period, between 2.5 and
17.5 days, the average pressure and temperatures are $(1.68\pm 0.02)$ bar and
$(-98.5\pm 0.02)$ ∘C, respectively. The run is terminated when the xenon gets
cryo-pumped into the storage bottle (3). This causes a pressure decrease down
to an adjusted gas amount of 1 bar inside the cryostat. Xenon gas is
recirculated in the gas system to speed up the warming up process (4).

##  5 DAQ, data processing and data analysis

###  5.1 Data acquisition system

The data acquisition system (DAQ) takes care of digitizing PMT waveforms. The
signals are taken directly from the HeXe signal feedthroughs to a custom-made
amplifier with an amplification factor of $\sim$2 per channel. They are then
fed into discriminators with a threshold at about –10 mV. The coincidence of
two discriminators above their threshold gives a trigger. Triggered data is
digitized with a CAEN V1724 module which has a 100 MHz sampling rate, 14 bit
resolution and a dynamic range of 2.25 V. A custom-made software is employed
to configure the module, to control data acquisition and to store the recorded
waveforms on disk. Figure 8 shows an example waveform of a
${}^{\mathrm{83m}}\mathrm{Kr}$ event in HeXe. The inset shows a zoom into the
S1 signal region with the signals corresponding to the two characteristic
decay lines.

Figure 8: ${}^{83\textrm{m}}$Kr candidate event. The signals and the intervals
as identified by the data processor are shown. The two main S2 signals are
followed by a small single-electron S2 signal which is caused by delayed
electron extraction or ionization of impurities inside the TPC.

On the right side, the signals from the two corresponding S2s can be seen. Due
to the large width of the S2s, most of the two S2s from the
${}^{\mathrm{83m}}\mathrm{Kr}$ decay do however overlap.

###  5.2 Data Processing

To process the acquired data, we use the software described in Cichon2020 . By
calculating the average of the first 25 available samples, the waveform
baseline is estimated. The baseline-subtracted waveforms from both PMTs are
added to a summed waveform which is used for analysis. The standard deviation
of the first 25 samples in the summed waveform, $\sigma$, is also computed.
Next, peak candidate regions are found by looking for excursions which surpass
$5\sigma$. Afterwards, a threshold-based peak finding algorithm, which is
detailed in Cichon2020 , is applied. For each peak found, its type — S1 or S2
— needs to be determined. Various pulse parameters, such as rise-time or pulse
width, are employed for this purpose. For S1 pulses, we require that the rise-
time, defined as the time between the 0.1 and 0.5 area quantiles, is below 200
ns and the pulse width (time duration between 0.1 and 0.9 area quantiles) to
be smaller than 500 ns. Peaks which do not satisfy these criteria are
classified as S2 pulses. The 0.5 area quantile is taken as the peak’s time
relative to the beginning of the event in which it is found.

Using the threshold-based peak splitting, an excellent separation between the
two S1s is achieved. S1s as close as 50 ns from each other can be resolved as
shown in figure 9. The figure shows the number of ${}^{83\textrm{m}}$Kr events
with two separated peaks for different decay times. For times between the S1s
smaller than about 125 ns, the efficiency of separating both lines decreases.
An exponential probability density function is fit to the distribution of time
differences between both S1s in order to determine the half-life of the
intermediate 9.4 keV state. This is done by minimizing the unbinned negative
log-likelihood in the range between 0.2 µs and 2 µs. The systematic error is
estimated by varying the lower fit bound by $\pm 75\,\mathrm{ns}$. From the
fit, a value of $T_{1/2}=\left(154.0\pm
0.6\,_{\mathrm{stat}}\,^{+0.4}{}_{-0.2}\,{}_{\mathrm{sys}}\right)\,\mathrm{ns}$
is extracted which is compatible with previous publications Ahmad:1995zz .

Figure 9: Time difference between the scintillation pulses (S1s) originating
from the decays of ${}^{83\textrm{m}}$Kr at 32.1 keV and 9.4 keV.

The PMT gain is defined as the average charge generated per photoelectron (PE)
in units of the elementary charge e-. In each run, the PMT gains are
calibrated using photons from a 350 nm LED located on trigger boards outside
the LXe chamber. Two optical fibers feed the light into the active volume of
the TPC, 20 mm below the gate. The same model-independent method as described
in Saldanha:2016mkn  is employed to regularly determine the gain of the PMTs.
Figure 10 shows the results of the gain calibration for the top and bottom
PMT,

Figure 10: Stability of the top and bottom PMT gains for two different HV
configurations. The dashed lines are constant fits to the data.

each at two different HV configurations for the period between June 2019 and
March 2020, with the dashed lines showing the fit of the data with a constant.
To process the data, gain values measured right before the respective
measurement are taken.

###  5.3 Data analysis

Alpha events from the 222Rn source are selected if precisely one S1 signal,
followed by an S2 signal are found in the waveform. For ${}^{83\textrm{m}}$Kr
events, we require that an S1 signal is followed by a smaller S1 and that at
least one S2 signal occurs in time after the larger S1 signal. Additionally,
the S2 width needs to fit the expected dependence with drift time, and the
fraction of the S1 signals seen in the top PMT needs to correspond to the
region of physical events.

The distribution of S2 photons among the two PMTs is used to select high
quality events as well. Due to an inclined Teflon piece at high radii on the
top of our TPC (see top panel of figure 5), the S2 area fraction seen by the
top PMT has a characteristic double peak structure that can be used to
effectively select an inner radius for analysis. A fiducial volume is defined
by cutting in the $z$-coordinate events such that
$10\,\textrm{mm}<z<35\,\textrm{mm}$. Especially for the two lowest fields
investigated, the data quality improves with time due to an unknown reason.
The statistics at the alpha line grows and the cathode position (see figure
18) becomes sharper. For this reason, we take only the second half of the
acquired data for the analysis of all fields below 60 V/cm. Variations on this
selection are performed to quantify the related systematic error.

The size of the recorded S1 and S2 signals depends on the event location in
the detector due to position-dependent effects, such as light collection
efficiency (LCE) and electron attachment to impurities in the liquid xenon
(see for Aprile:2012vw  details). The signal sizes are corrected for the
$z$-dependence of these effects. Two different corrections are necessary as
the attenuators in the 222Rn measurements modify the light collection of each
PMT differently than in ${}^{83\textrm{m}}$Kr. The selection of
${}^{83\textrm{m}}$Kr events for a drift field of 394 kV/cm is shown as an
example in figure 11 (region above the red lines).

Figure 11: Population of ${}^{83\textrm{m}}$Kr events and the selection (red
lines) using the time coincidence of its 32.1 keV and 9.4 keV decay lines.

Figure 12 shows the dependence of the uncorrected S1 light yield on the drift
time due to light collection effects.

Figure 12: Dependence of the 32.1 keV S1 from ${}^{83\textrm{m}}$Kr on the
drift time due to the geometric variation of the light collection efficiency.

The drift time dependence of the charge signal, where the signal loss is due
to attachment of electrons to electronegative impurities in the liquid, can be
seen in the lower right panel of figure 5. It follows an exponential decay
law, where the time-constant is usually referred to as the electron lifetime.
To evaluate the $z$-dependence of the S2 signal size, the median value of the
event distribution is determined for different slices along the
$z$-coordinate. These values are then fit with an exponential function which
is used to correct the data to the value at the top of the TPC where no losses
are expected. For the datasets employed for the charge and light yield
measurements, an average electron lifetime is used for the correction of each
field. This is enough as each field is measured for just 4 hours. Note that
while the S1 $z$-correction can be determined once for the complete run, the
S2 $z$-dependence is time dependent due to the continuous cleaning of the
liquid xenon which results in a constantly improving S2 collection.

Alpha events are selected by looking at the highest energies in the recorded
spectrum and by considering the known position in the S2 versus S1 space.
Therefore, only a coarse energy cut on S1 and S2 is used to select those
events. Figure 13 shows the 222Rn data in this parameter space for an electric
field of $1.24$ kV/cm.

Figure 13: 222Rn data in the space of corrected S1 versus S2 at an electric
drift-field of 1.24 kV/cm. The estimated mean S1 and S2 signal sizes of the
three isotopes are indicated by the pink markers.The contour lines of the fit
(sum of three individual 2D-Gaussian functions) are drawn in black. The dashed
red line shows the linear fit through the fitted mean positions.

Three different decay lines can be identified originating from 222Rn, 218Po
and 214Po at energies of 5.5, 6.0 and 7.7 MeV, respectively. These three lines
are only separable for electric fields above 70 V/cm. For lower fields, the S2
signal size is significantly reduced due to an increased probability of
electron-ion recombination in the dense tracks created by alpha particles,
making the different decay lines inseparable within our S2 resolution. A
projection of the alpha lines onto the S2 axis can be seen in figure 14. With
the alpha decay energies of 222Rn and 218Po being very close, the two
corresponding peaks clearly overlap. Their positions however can still be
fitted separately to extract light and charge yield for both lines.

Figure 14: Corrected charge signal (S2) spectrum of the alpha decays from the
222Rn-calibration at an electric drift field of 1.24 kV/cm.

The energy resolution, defined as the ratio between the standard deviation and
the mean of the fitted Gaussian, for alpha interactions is found to be around
5% both for S1 and S2 signals at 1.2 kV/cm. While the S1 resolution is almost
constant for varying drift fields, the S2 resolution worsens slightly with
decreasing electric field due to the lower charge collection. For the
${}^{83\textrm{m}}$Kr signals, we obtain 29% and 20% resolution for the 9.4
and 32.1 keV S1 lines and 19% for the combined 41.5 keV S2 line.

##  6 Response of liquid xenon to electrons and $\mathbf{\upalpha}$-particles

The creation of scintillation and ionization signals in LXe is a complex
process which has been investigated intensively over the past decades
Aprile:2006kx ; Aprile:2007qd ; Manalaysay:2009yq ; Lin:2013ypa ;
Hogenbirk:2018knr ; Temples:2021jym . In the absence of an electric field,
ionization electrons recombine with the positively charged ions, leading to
additional S1 scintillation photons. In the presence of an applied field, the
electrons are drifted out of the interaction site to be read out separately.
The amount of charge and light depend on the electric field, i.e. the higher
the field, the more efficiently electrons are prevented from recombining,
leading to a higher charge yield and a lower light yield. For high enough
fields, the charge and light yields reach a plateau.

Alpha particles have a high dE/dx and therefore the ionization cloud is
significantly denser than for electrons. As a result, most electrons are
screened by the surrounding ionized medium and cannot escape recombination.
The light yield is consequently higher and the charge yield lower than for an
electron of equivalent energy. In this section, we investigate the signal
production for low energy electrons and alpha particles using the data taken
with the ${}^{\mathrm{83m}}\mathrm{Kr}$ and 222Rn sources.

First, we parameterize the S1-S2 anti-correlation in our TPC using the
measurements performed at various drift fields. Figure 15 shows the corrected
S1 and S2 values for different particles.

Figure 15: Anti-correlation of charge and light for the 41.5 keV line of the
${}^{83\textrm{m}}$Kr source (red) and for the alpha particles of 222Rn and
218Po in lower left inset. Only data points with a drift field above 70 V/cm
are shown as only those are used for the fit. For better visibility, the data
points of 214Po are not shown in the inset.

In the case of ${}^{\mathrm{83m}}\mathrm{Kr}$, the light yield of the two
transitions at 9.4 keV and 32.1 keV can be measured separately. However, due
to the short half-life of the intermediate state, the S2 signals overlap and
only a combined measurement (41.5 keV) is possible for the charge yield. For
this reason, the main panel of figure 15 contains only the 41.5 keV data. The
observed anti-correlation is then fit with a linear function in order to
determine values for $L_{0}$ and $Q_{0}$, which are needed for the
normalization of charge and light yield. Here $L_{0}$ is defined as the
theoretical light yield for the case of vanishing ionization, whereas $Q_{0}$
is the theoretical amount of charge when there would be no excitation.

The 222Rn data is shown in an inset. Note that the detected S1 and S2 signals
are not comparable between the two datasets due to the presence of attenuators
in the 222Rn case which modifies the light collection. For the 222Rn data, we
exclude the data points below 70 V/cm from figure 15, as the data quality is
significantly better for drift fields above that. Overall data points are very
clustered (due to the small quenching of alpha particles) and the fit has a
significant uncertainty. This uncertainty is propagated as systematic error
and it is represented in figure 17 by gray error bars. The data points for
222Rn and 218Po are shifted from each other (approx. 1%), likely due to the
different dE/dx of the two alpha lines.

Figure 16 shows the field dependence of the normalized charge and light
signals for the ${}^{\mathrm{83m}}\mathrm{Kr}$ decay lines.

Figure 16: Light and charge yields of the ${}^{\mathrm{83m}}\mathrm{Kr}$
transitions as function of applied electric field $E$, measured using the HeXe
setup (circles). The data points from Manalaysay:2009yq  (crosses) are also
shown. The dotted lines show a fit with the modified Thomas-Imel box model
given in equation 1 (data points are provided online HeXeData ).

For the combined 41.5 keV S2 line (red), the normalization $Q_{0}$ was
determined from figure 15, following the approach described in
Manalaysay:2009yq . The normalization factors $L_{0}$ of the light signals
from the 9.4 keV and 32.1 keV emission lines (purple and orange) are
determined via extrapolation to the light signal at zero field. To avoid the
variation of the 9.4 keV signal yield with the time between the 32.1 keV and
the 9.4 keV S1 signals as seen in Baudis:2013cca , we select events in which
the two corresponding S1 signals are more than 300 ns apart. We find a good
agreement between our data and the phenomenological function (black dashed
line) used in Manalaysay:2009yq , which is based on the Thomas-Imel box model
Thomas:1987zz , expressed as:

$\displaystyle\frac{L(E)}{L_{0}},\frac{Q(E)}{Q_{0}}=a_{1}\cdot a_{2}\cdot
E\cdot\ln{\left(1+\frac{1}{a_{2}\cdot E}\right)}+a_{3}\,.$ (1)

$E$ is the applied electric field, $L(E)$ and $Q(E)$ are the light and charge
yields with their normalization constants $L_{0}$ and $Q_{0}$, and $a_{1}$,
$a_{2}$ and $a_{3}$ are fit parameters. Their results and statistical
uncertainties are reported in table 1.

Table 1: Best fit values for the field dependent light and charge yield variation for ${}^{\mathrm{83m}}\mathrm{Kr}$ (shown in figure 16) explained by the modified Thomas-Imel box model shown in equation 1. Only statistical uncertainties are reported. Transition | $a_{1}$ | $a_{2}\,\mathrm{\left[cm/kV\right]}$ | $a_{3}$
---|---|---|---
9.4 keV | $-1.0\,\pm\,0.3$ | $0.06\,\pm\,0.03$ | 1 (fixed)
32.1 keV | $-0.443\,\pm\,0.011$ | $1.41\,\pm\,0.10$ | 1 (fixed)
41.5 keV | $0.375\,\pm\,0.008$ | $1.11\,\pm\,0.11$ | $0.091\,\pm\,0.004$

The light yield results are in agreement with the data in Manalaysay:2009yq
for drift fields above 0.4 kV/cm. Below this value, the shape differs
slightly. The charge yield of the summed signal at 41.5 keV shows a shape
similar to Manalaysay:2009yq  but our values are systematically lower. Note
that our data include several measurements at drift fields below 200 V/cm
which were never measured systematically before.

The light and charge yields of $\upalpha$-particles from the 222Rn source are
measured in a new run because Teflon attenuators need to be introduced to
prevent signal saturation. For fields above 70 V/cm, we fit the data with a
sum of three individual 2D Gaussian functions (see figure 13) obtaining light
and charge yields of 222Rn, 218Po, and 214Po separately. For lower fields the
$\upalpha$-decays from 222Rn and 218Po become indistinguishable and therefore,
only a combined fit is possible. The 222Rn light yield values are corrected
taking into account the bias produced by the fitting of two lines together.
Figure 17 shows the resulting light and charge yields as function of electric
field for alpha particles.

Figure 17: Light and charge yields for the $\upalpha$-decays of 222Rn, 218Po,
and 214Po as function of electric field. The data is compared with the results
from Aprile:1991xb  and Aprile:2006kx . Gray bars indicate the systematic
error. Datapoints are provided online HeXeData .

The uncertainty on the field strength (too small to be visible in the figure)
is calculated from the electric field simulation as the width of the central
68th percentile of the electric field values throughout the fiducial volume.
The vertical error bars include a statistical error from the number of events
as well as a systematic error (gray). The latter is estimated by varying
analysis parameters like selection cuts or the fiducial volume and it is
combined with the uncertainty due to the normalization method described above.

The data is shown together with existing data from Aprile et al. Aprile:1991xb
; Aprile:2006kx . While the light yield measurements agree with each other
within the uncertainty, it can be seen that for our measurements at fields
below $\sim 70$ V/cm the normalized light yields exceed values of 1. We
attribute this to field inhomogeneities due to field leakage from the
extraction region into the drift region at lower fields (see figure 4). This
in turn causes the S1 light collection correction to become less effective and
therefore these data points are excluded from the normalization shown in
figure 15. The curve of the charge yield is found to be significantly lower as
compared to the values reported in Aprile:1991xb ; Aprile:2006kx  and flattens
slightly above 1.3 kV/cm. We discuss in the following possible explanations
for the discrepancy based on both different experimental conditions and the
different data analysis.

The reported liquid xenon temperature in Aprile:1991xb  is 195 K, which is
approximately 20 K higher compared to our measurement. Consequently, the
liquid xenon density in our measurement was higher by about 5.1% AirLiquid .
This should lead to a slightly higher ionization density as well, resulting in
a lower charge yield. Moreover, while we use a homogeneously distributed
source, for the measurements reported in Aprile:1991xb ; Aprile:2006kx  the
source was directly deposited onto the cathode surface. This restricts the
detectable alpha emission angles to the 2$\uppi$ upward direction, whereas in
our measurement alpha particles emitted under all possible angles are
detectable. Horizontal $\upalpha$-tracks are expected to have a slightly
higher charge yield compared to tracks which are aligned with the field (see
also discussion in Aprile:1991xb ). For vertically aligned tracks, electrons
need to drift through the volume of the track, where a high ion-density is
present, increasing their probability to recombine. Our measurement contains
all 4$\uppi$ possible directions of the alpha particle which could also
contribute to the observed difference.

Furthermore, a small fraction of the alpha decay energy is carried away by the
recoil of the daughter nucleus. In our measurement, we observe the sum of the
alpha-particle energy and the $\mathcal{O}(100\,\textrm{keV})$ recoil of the
daughter nucleus, while in Aprile:1991xb ; Aprile:2006kx  this recoil is
absorbed by the cathode and therefore not detectable. In our data, we can make
use of a fiducial volume selection, to remove events close to the electrodes
avoiding the strongly distorted fields in those regions. Using our detailed 3D
detector simulation shown in section 3, we have a good knowledge of the
electric field and its uncertainty within our active volume. The field
strength close to the cathode surface in Aprile:1991xb ; Aprile:2006kx  might
be enhanced due to surface irregularities. This possibility was considered and
mitigated by polishing the cathode prior to the deposition of the sources.
Nevertheless, this could contribute slightly to the discrepancy.

The normalization of the measured charge yield by the value of $Q_{0}$ was
also performed differently in both measurements. For our data, we employed the
method illustrated in figure 15 which makes use of the simultaneous
measurement of charge and light signal and their anti-correlation. In
Aprile:1991xb  the normalization factor is determined using the ratio between
the alpha decay energy and the average energy $W=15.6$ eV required to produce
an electron-ion pair in liquid xenon. Note, however, that this assumed value
is also significantly larger than the ones reported in more recent
measurements $W=11.5$ eV EXO-200:2019bbx ; Baudis:2021dsq . Using this value,
the charge yield values of Aprile should decrease by approximately 25%,
reducing the discrepancy with our data.

##  7 Drift velocity measurement

In liquid xenon TPCs, the $z$-coordinate or depth of an event is reconstructed
from the time difference between the prompt scintillation light (S1) and the
proportional light from the drifted electrons (S2). The conversion from time
to distance can be easily performed using the drift velocity which is usually
constant when the electric field is sufficiently homogeneous. This section
describes a measurement of the electron drift velocity in liquid xenon at a
temperature of $(174.4\pm 0.2)$ K for electric field values between 7.5 and 1
642 V/cm.

To extract the drift velocity, we divide the distance between gate and
cathode, which amounts to $\mathrm{(49.3\pm 0.3)\,mm}$ at the liquid xenon
temperature, by the time that the electrons take to travel between these
electrodes. The time is extracted from the features that alpha-particle energy
depositions produce at the gate and the cathode. Since alpha particles produce
very bright S1 signals, electrons are released from the stainless steel
surfaces of the meshes via the photoelectric effect. This happens in time
coincidence with the S1 signal. The emitted electrons are drifted to the
amplification region, where they produce small S2 signals. Due to the fixed
drift distances, the positions of the electrodes can be recognized in the data
as peaked structures. Figure 18 shows the drift time distribution of events
with S2 signals smaller than 25 PE zooming into the gate region (left inset)
and the cathode region (right inset).

Figure 18: Drift time distribution of small-sized S2 events for 222Rn data in
the TPC for a field of 225 V/cm zooming into the gate region (left inset) and
the cathode region (right inset).

The position of the gate is determined by fitting a Gaussian function to the
peak. The cathode position is fit using a Gaussian function for fields larger
than 17 V/cm and a right-tailed Crystal-ball function for lower drift-fields
(red lines in figure 18). For each drift-field configuration, both fits are
repeated 20 times, where the fit region is varied within $[\pm
1\sigma,\ldots\pm 2\sigma]$. The maximum variation between the fitted mean
positions is then added to the uncertainty from the geometric distance.

The drift velocity of electrons can also be determined using the
${}^{83\textrm{m}}$Kr data acquired at an operational temperature of
$(173.41\pm 0.11)$ K in a drift field range between 13.5 V/cm to 1 245 V/cm.
In this case, the positions of gate and cathode are determined in a different
way. The charge yield in the extraction region typically differs from the one
in the drift volume. The change in the S2 size of ${}^{83\textrm{m}}$Kr can be
therefore used to determine the position of the gate. The position of the
cathode coincides with a steep decrease of the event rate, since no electrons
can be drifted upwards from the region below the cathode. We determined the
cathode position as the time where this decrease is maximal, following a
similar approach as reported in Baudis:2017xov .

The derived drift velocity values from the alpha data are displayed in figure
19.

Figure 19: Drift velocity of electrons in LXe from alpha-particle interactions
as measured using the HeXe setup at T$=174\,\mathrm{K}$ as a function of
applied drift field. Horizontal gray bars indicate the central 68% quantile
range of the field values throughout the drift volume. For better visibility
the data points from ${}^{83\textrm{m}}$Kr are not shown but are provided
online HeXeData .

For the x-axis, the vertical component of the electric field is taken as this
is what determines the electron drift. For most of the data, this vertical
component corresponds to the absolute value of the electric field, except for
very low fields for which the absolute field is higher due to field
distortions. Most data points were measured twice, however, they are not
separately visible in the figure because they lie exactly on top of each
other. The only exception is the point at 13 V/cm, where in one of the two
datasets the data quality was significantly poorer for unknown reasons, likely
due to charge up following a HV trip. Data from the EXO-200 Albert:2016bhh ,
XENON100 Aprile:2011dd  and XENON1T XENON:2019izt  experiments are displayed
together with the measurements of dedicated setups Baudis:2017xov ;
Hogenbirk:2018knr ; Miller:1968zza ; Gushchin:1982b ; Njoya:2019ldm . The
drift velocity derived using ${}^{83\textrm{m}}$Kr events are in agreement
with the ones in the figure within errors.

Our data covers a large range in electric fields from 7.5 up to 1 642 V/cm and
is overall in good agreement with literature data. The systematic
discrepancies with some of the data sets are attributed to the different
operating temperatures as argued in Baudis:2017xov . The data from Baudis for
instance shows lower drift velocity values due to the higher operating
temperature of 184 K. Following the same argument, the drift velocity values
from Miller and EXO are larger as they were acquired at 163 K and 167 K,
respectively (see also figure 20).

We fit the data with a model based on the phenomenological approach of the
NEST Szydagis_2011  framework using a sum of individual exponential functions
and an offset to describe the dependence of drift velocity $v_{d}$ to the
applied drift field $E$. We find that the data within the investigated range
of drift fields can be sufficiently described using only two exponential
functions:

$\displaystyle v_{d}(E)=A_{1}\cdot e^{-E/B_{1}}+A_{2}\cdot e^{-E/B_{2}}+C\,.$
(2)

The constants $A_{1,2}$, $B_{1,2}$ and $C$ are determined from data for a
given LXe temperature. Table 2 summarizes the found values for the two
calibration sources ${}^{83\textrm{m}}$Kr as well as 222Rn.

Table 2: Summary of fit parameters to equation 2 for ${}^{83\textrm{m}}$Kr and 222Rn measurements at given temperatures Parameter |  ${}^{83\textrm{m}}$Kr |  222Rn | Unit
---|---|---|---
Temperature | $173.41\,\pm\,0.11$ | $174.4\,\pm\,0.2$ | K
$A_{1}$ | $-1.38\,\pm\,0.02$ | $-1.458\,\pm\,0.013$ |
$A_{2}$ | $-0.95\,\pm\,0.13$ | $-0.95\,\pm\,0.02$ | $\mathrm{mm/\upmu s}$
$B_{1}$ | $38.0\,\pm\,1.8$ | $34.7\,\pm\,1.2$ |
$B_{2}$ | $1000\,\pm\,300$ | $830\,\pm\,70$ | $\mathrm{kV/cm}$
$C$ | $2.33\,\pm\,0.15$ | $2.27\,\pm\,0.04$ | $\mathrm{mm/\upmu s}$

The dependence of the drift velocity on the LXe temperature is shown in figure
20 for a range of drift fields between 0.1 and 1 kV/cm where several
measurements exist.

Figure 20: Dependence of the drift velocity on the temperature for a few
selected values of the drift field.

A subset of data sets from figure 19 is employed for which the measurement
temperature was clearly stated. Data at 162 K Njoya:2019ldm , 165 K
Gushchin:1982b , 167 K Albert:2016bhh , 173.4 K and 174.4 K (this study) and
184 K Baudis:2017xov  is used. The values for the drift velocity at each
electric field are determined interpolating linearly in between the published
data when necessary. The electron mobility depends on temperature roughly as
$\mu_{e}\propto T^{-3/2}$ Miller:1968zza . Using the dependence of the drift
velocity on the electric field, $v_{d}=\mu\cdot E$, the data of figure 19
could in principle be fit with a $\propto T^{-3/2}$ dependence. However, this
functional dependence is unable to describe the steep increase of drift
velocity at cold temperatures for high fields ($E>0.6$ kV/cm). Therefore and
just to guide the eye, we fit a linear function to the points of each electric
field. Our data points are acquired at a temperature close to the expected
operating temperature of XENONnT and therefore are of interest for the
experiment.

##  8 Conclusions

In this work, we describe the performance of our small-scale liquid-xenon TPC,
HeXe, which is a multipurpose R&D detector. A new 3 dimensional electric field
simulation using COMSOL Multiphysics® allows us to optimize the design for a
uniform electric field and to quantify the error on the field which is often
ignored. This is especially important for the measurements performed at fields
below $\sim 100$ V/cm.

We employ data from a ${}^{83\textrm{m}}$Kr and a 222Rn source to characterize
the light and charge yields of the corresponding decay lines. For
${}^{83\textrm{m}}$Kr, we systematically investigate low drift fields down to
13.5 V/cm where no measurements existed in the literature so far. We study the
signal yields of the three $\upalpha$-lines 222Rn, 218Po and 214Po. We find a
significant deviation of the charge yield compared to existing data. Although
the origin of the discrepancy is not understood, we give several hypotheses
including different analysis methods but also different experimental
conditions.

The drift velocity of ionization electrons is measured for field values
between 7.5 and 1 640 V/cm at a LXe temperature of 174.4 K. Several
measurements at fields below 100 V/cm were taken to characterize the low field
region for which literature is scarce. We study the temperature dependence of
the drift velocity using also literature values and find an increase in drift
velocity for decreasing temperatures which is most pronounced at high fields.

###### Acknowledgements.

We acknowledge the support of the the Max Planck Society. We thank our
technicians Jonas Westermann, Steffen Form, and Michael Reißfelder for their
technical support during construction and operation of the system.
Furthermore, we thank our colleges Hardy Simgen, Stefan Brünner and Sebastian
Lindemann for their contributions and very fruitful discussions.

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