∎

email_fte-mail<EMAIL_ADDRESS>email_sle-mail:
<EMAIL_ADDRESS>email_mke-mail:
<EMAIL_ADDRESS>deceasedDeceased alsoatromaAlso at
INFN - Roma Tre, 00146 Roma, Italy alsoatcoimbrapoliAlso at Coimbra
Polytechnic - ISEC, 3030-199 Coimbra, Portugal alsoatuniheidelbergAlso at
Physikalisches Institut, Universität Heidelberg, Heidelberg, Germany

11institutetext: Physics Department, Columbia University, New York, NY 10027,
USA 22institutetext: Kamioka Observatory, Institute for Cosmic Ray Research,
and Kavli Institute for the Physics and Mathematics of the Universe (WPI),
University of Tokyo, Higashi-Mozumi, Kamioka, Hida, Gifu 506-1205, Japan
33institutetext: LPNHE, Sorbonne Université, CNRS/IN2P3, 75005 Paris, France
44institutetext: Institut für Kernphysik, Westfälische Wilhelms-Universität
Münster, 48149 Münster, Germany 55institutetext: INAF-Astrophysical
Observatory of Torino, Department of Physics, University of Torino and INFN-
Torino, 10125 Torino, Italy 66institutetext: Nikhef and the University of
Amsterdam, Science Park, 1098XG Amsterdam, Netherlands 77institutetext: Oskar
Klein Centre, Department of Physics, Stockholm University, AlbaNova, Stockholm
SE-10691, Sweden 88institutetext: Department of Physics & Kavli Institute for
Cosmological Physics, University of Chicago, Chicago, IL 60637, USA
99institutetext: New York University Abu Dhabi - Center for Astro, Particle
and Planetary Physics, Abu Dhabi, United Arab Emirates 1010institutetext:
Physik-Institut, University of Zürich, 8057 Zürich, Switzerland
1111institutetext: Department of Physics and Astronomy, Purdue University,
West Lafayette, IN 47907, USA 1212institutetext: SUBATECH, IMT Atlantique,
CNRS/IN2P3, Université de Nantes, Nantes 44307, France 1313institutetext:
Department of Physics and Astronomy, University of Bologna and INFN-Bologna,
40126 Bologna, Italy 1414institutetext: Max-Planck-Institut für Kernphysik,
69117 Heidelberg, Germany 1515institutetext: Physikalisches Institut,
Universität Freiburg, 79104 Freiburg, Germany 1616institutetext: Department of
Particle Physics and Astrophysics, Weizmann Institute of Science, Rehovot
7610001, Israel 1717institutetext: Department of Physics & Center for High
Energy Physics, Tsinghua University, Beijing 100084, China 1818institutetext:
LIBPhys, Department of Physics, University of Coimbra, 3004-516 Coimbra,
Portugal 1919institutetext: INFN-Laboratori Nazionali del Gran Sasso and Gran
Sasso Science Institute, 67100 L’Aquila, Italy 2020institutetext: Institute
for Astroparticle Physics, Karlsruhe Institute of Technology, 76021 Karlsruhe,
Germany 2121institutetext: Department of Physics and Astronomy, Rice
University, Houston, TX 77005, USA 2222institutetext: Department of Physics
and Chemistry, University of L’Aquila, 67100 L’Aquila, Italy
2323institutetext: Institut für Physik & Exzellenzcluster PRISMA+, Johannes
Gutenberg-Universität Mainz, 55099 Mainz, Germany 2424institutetext:
Department of Physics, University of California San Diego, La Jolla, CA 92093,
USA 2525institutetext: Department of Physics “Ettore Pancini”, University of
Napoli and INFN-Napoli, 80126 Napoli, Italy 2626institutetext: Kobayashi-
Maskawa Institute for the Origin of Particles and the Universe, and Institute
for Space-Earth Environmental Research, Nagoya University, Furo-cho, Chikusa-
ku, Nagoya, Aichi 464-8602, Japan 2727institutetext: Department of Physics,
Kobe University, Kobe, Hyogo 657-8501, Japan 2828institutetext: INFN - Ferrara
and Dip. di Fisica e Scienze della Terra, Università di Ferrara, 44122
Ferrara, Italy

# Design and performance of the field cage for the XENONnT experiment

E. Aprilecolumbia  K. Abetokyo  S. Ahmed Maouloudparis  L. Althuesermunster
B. Andrieuparis  E. Angelinotorino  J. R. Angevaarenikhef  V. C.
Antochistockholm  D. Antón Martinchicago  F. Arneodonyuad  L. Baudiszurich  A.
L. Baxterpurdue  M. Bazyksubatech  L. Bellagambabologna  R. Biondiheidelberg
A. Bismarkzurich  E. J. Brookesnikhef  A. Brownfreiburg  S. Bruennernikhef  G.
Brunosubatech  R. Budnikwis  T. K. Buitokyo  C. Caitsinghua  J. M. R.
Cardosocoimbra  D. Cichonheidelberg  A. P. Cimental Chávezzurich  A. P.
Colijnnikhef  J. Conradstockholm  J. J. Cuenca-Garcíazurich  J. P.
Cussonneausubatech,deceased  V. D’Andrealngs,alsoatroma  M. P. Decowskinikhef
P. Di Gangibologna  S. Digliosubatech  K. Eitelkit  A. Elykovkit  S.
Farrellrice  A. D. Ferellalaquila,lngs  C. Ferrarilngs  H. Fischerfreiburg  M.
Fliermannikhef  W. Fulgionetorino,lngs  C. Fusellinikhef  P. Gaemersnikhef  R.
Gaiorparis  A. Gallo Rossostockholm  M. Gallowayzurich  F. Gaotsinghua  R.
Glade-Beuckefreiburg  L. Grandichicago  J. Grigatfreiburg  H. Guanpurdue  M.
Guidaheidelberg  R. Hammannheidelberg  A. Higuerarice  C. Hilsmainz  L.
Hoetzschheidelberg  N. F. Hooducsd  J. Howlettcolumbia  M. Iacovaccinapels  Y.
Itownagoya  J. Jakobmunster  F. Joergheidelberg  A. Joystockholm  M. Karakit
P. Kavriginwis  S. Kazamanagoya  M. Kobayashinagoya,email_mk  G. Koltmanwis
A. Kopecucsd  F. Kugerfreiburg  H. Landsmanwis  R. F. Langpurdue  L.
Levinsonwis  I. Lirice  S. Lipurdue  S. Liangrice  S.
Lindemannfreiburg,email_sl  M. Lindnerheidelberg  K. Liutsinghua  J.
Loizeausubatech  F. Lombardimainz  J. Longchicago  J. A. M.
Lopescoimbra,alsoatcoimbrapoli  Y. Maucsd  C. Macolinolaquila,lngs  J.
Mahlstedtstockholm  A. Mancusobologna  L. Manentinyuad  F. Marignettinapels
T. Marrodán Undagoitiaheidelberg  K. Martenstokyo  J. Masbousubatech  D.
Massonfreiburg  E. Massonparis  S. Mastroianninapels  M. Messinalngs  K.
Miuchikobe  A. Molinariotorino  S. Moriyamatokyo  K. Moråcolumbia  Y.
Mosbacherwis  M. Murracolumbia  J. Müllerfreiburg  K. Niucsd  U. Oberlackmainz
B. Paetschwis  J. Palacioheidelberg  Q. Pellegriniparis  R. Pereszurich  C.
Petersrice  J. Pienaarchicago  M. Pierrenikhef  G. Plantecolumbia  T. R.
Pollmannnikhef  J. Qiucsd  J. Qinpurdue  D. Ramírez Garcíazurich  N.
Šarčevićfreiburg  J. Shitsinghua  R. Singhpurdue  L. Sanchezrice  J. M. F. dos
Santoscoimbra  I. Sarnoffnyuad  G. Sartorellibologna  J. Schreinerheidelberg
D. Schultemunster  P. Schultemunster  H. Schulze Eißingmunster  M.
Schumannfreiburg  L. Scotto Lavinaparis  M. Selvibologna  F. Semeriabologna
P. Shaginmainz  S. Shicolumbia  E. Shockleyucsd  M. Silvacoimbra  H.
Simgenheidelberg  A. Takedatokyo  P.-L. Tanstockholm  A.
Terliukheidelberg,alsoatuniheidelberg  D. Therssubatech  F. Toschikit,email_ft
G. Trincherotorino  C. Tunnellrice  F. Tönniesfreiburg  K. Valeriuskit  G.
Voltazurich  C. Weinheimermunster  M. Weisswis  D. Wenzmainz  C. Wittwegzurich
T. Wolfheidelberg  V. H. S. Wukit  Y. Xingsubatech  D. Xucolumbia  Z.
Xucolumbia  M. Yamashitatokyo  L. Yangucsd  J. Yecolumbia  L. Yuanchicago  G.
Zavattiniferrara  M. Zhongucsd  T. Zhucolumbia

(Received: date / Accepted: date)

###### Abstract

The precision in reconstructing events detected in a dual-phase time
projection chamber depends on an homogeneous and well understood electric
field within the liquid target. In the XENONnT TPC the field homogeneity is
achieved through a double-array field cage, consisting of two nested arrays of
field shaping rings connected by an easily accessible resistor chain. Rather
than being connected to the gate electrode, the topmost field shaping ring is
independently biased, adding a degree of freedom to tune the electric field
during operation. Two-dimensional finite element simulations were used to
optimize the field cage, as well as its operation. Simulation results were
compared to ^83mKr calibration data. This comparison indicates an accumulation
of charge on the panels of the TPC which is constant over time, as no
evolution of the reconstructed position distribution of events is observed.
The simulated electric field was then used to correct the charge signal for
the field dependence of the charge yield. This correction resolves the
inconsistent measurement of the drift electron lifetime when using different
calibrations sources and different field cage tuning voltages.

††journal: Eur. Phys. J. C

##  1 Introduction

The strongest direct constraints on dark matter in the form of weakly
interacting massive particles (WIMPs) come from noble liquid-gas dual-phase
time projection chambers (TPCs) XENON:2023sxq ; xenon1t_results ; lz_results ;
pandax_results ; darkside_results ; cevns . The XENONnT experiment, located at
the INFN Laboratori Nazionali del Gran Sasso (LNGS) in central Italy, deploys
a dual-phase TPC with a liquid xenon (LXe) target of
$5.9\text{\,}\mathrm{t}\text{/}$ and set an upper limit on the spin-
independent WIMP-nucleon elastic scattering cross section down to
$2.58\text{\times}{10}^{-47}\text{\,}\mathrm{{}^{2}}{cm}$ for a
$28\text{\,}\mathrm{G}\mathrm{eV}\mathrm{/}\mathrm{{}^{2}}\mathrm{c}$ WIMP
mass at $90\text{\,}\mathrm{\char 37\relax}\text{/}$ confidence level
XENON:2023sxq .

A particle interacting in the liquid xenon target produces a prompt
scintillation light signal (S1) and frees ionization electrons. The S1 vacuum-
ultraviolet (VUV) photons are detected by a top and a bottom array of
photomultiplier tubes (PMTs), while the electrons drift upwards following the
electric drift field created by a cathode and a gate electrode. They are then
accelerated into a high electric field region between gate and anode. There
they are extracted into the gaseous phase and produce a secondary proportional
scintillation signal (S2) before being collected on the anode electrode. The
localized nature of the S2 signal allows an $\left(x,y\right)$-position
reconstruction based on the detected light distribution in the top PMT array,
while the time difference between the S1 and the S2 signal gives an estimate
for the $z$ coordinate. The ratio between S1 and S2 provides information about
the nature of the underlying interaction. For a given S1 signal, nuclear
recoils (NRs) of WIMP or neutron interactions are characterized by a smaller
S2 signal than electronic recoils (ERs) from beta or gamma interactions.

The electric field at the interaction point in the LXe also affects the signal
ratio S2/S1. For this reason, a homogeneous and well understood electric drift
field is crucial for a good discrimination between NR and ER events, and to
achieve the best sensitivity for a WIMP search. The electric fields of the
XENONnT TPC are produced by a set of five electrodes (anode, gate, cathode and
two screening electrodes) and a field cage enclosing the active volume. The
field cage consists of an inner and an outer array of concentric conductive
rings connected by two redundant resistor chains. A sketch of the TPC with the
position of the electrodes and the field cage is shown in Fig. 1.

Figure 1: Sketch of the XENONnT TPC. The zoom-in shows a detail of the double-
array structure of the field cage, whose implementation is shown in Fig. 2.

This paper focuses on the simulation and design of the field cage for the
XENONnT experiment, with particular emphasis on its improvements with respect
to the predecessor experiment, XENON1T XENON:2017lvq . The design and
implementation of the field cage are described in Sect. 2. The field
simulation setup and the optimization of the resistor chain are summarized in
Sect. 3, focusing on the freedom to tune the drift field by changing the
voltage applied to the topmost ring of the field cage, treated as an
independent electrode. Finally, the field cage tuning results of XENONnT are
discussed in Sect. 4. This section also shows the matching of data to
simulations of the XENONnT electric field during the first science run, which
includes a charge-up component on the TPC reflective walls.

##  2 The XENONnT field cage

A WIMP scattering in LXe is expected to produce a small scintillation signal
S1, hence it is crucial to maximize the light collection efficiency (LCE) of
the detector. In addition to the use of VUV-reflective polytetrafluorethylen
(PTFE) walls enclosing the full instrumented target yamashita_ptfe , XENONnT
deploys electrodes which are highly transparent to light. This was achieved
using a parallel wire grid design with an optical transparency exceeding
$95\text{\,}\mathrm{\char 37\relax}\text{/}$ nt_instrument_paper . The
electrodes need to sustain high voltages, as high electric drift fields are
known to reduce the fraction of ER events misclassified as NRs, improve the
discrimination power between single and multiple scatter, and reduce the
maximum electron drift time, limiting the accidental coincidence background
LUX:2020car . The design drift field of XENONnT was
$200\text{\,}\mathrm{V}\text{/}\mathrm{cm}$, aiming at a larger value than
achieved in XENON1T, while considering the past difficulties for dual-phase
TPCs in reaching high voltages at the cathode Rebel:2014uia .

The optical transparency of the electrodes translates into a significant field
leakage into the drift volume of the extraction field from above the gate and
of the reverse field from below the cathode. This results in an inhomogeneous
field within the active volume, leading to a spatially-dependent S2/S1 signal
ratio. This spatial dependence negatively impacts the discrimination power
between signal-like nuclear recoils and background-like electronic recoils,
ultimately affecting the final sensitivity to WIMPs LUX:2020car . The field
cage plays a crucial role in addressing the problem of field inhomogeneity,
forcing a constant voltage gradient within the active volume and effectively
mitigating the field leakage through the electrodes.

The field cage is composed of an inner and outer set of oxygen-free high
conductivity (OFHC, $99.99\text{\,}\mathrm{\char 37\relax}\text{/}$) copper
rings that are connected by a chain of resistors and enclose the entire length
of the TPC. It is positioned on the outside of the reflecting panels to
prevent scintillation photons from being lost due to the photoelectric effect
as they hit the copper rings of the field cage, which would reduce the LCE and
release single electrons. A section of the field cage, along with the chain of
resistors and various PTFE parts, is shown in Fig. 2.

Figure 2: View of the XENONnT field cage from the outside of the TPC during
assembly in the clean room. It is possible to discern the different elements:
the inner array rings (a) clipped in the notches (b) on the sliding PTFE
panels (c) and connected via the resistor chain (d). The outer rings array (e)
and its resistor chain (f) are also visible. The pillars (g) are still open as
this picture was taken during the assembly. During nominal operation, covers
(h) are placed to fix the outer rings. The indicated dimension is given at
liquid xenon temperature.

The double-array structure of the XENONnT field cage introduces a novel
approach in the field of dual-phase TPCs. The rigid outer rings act as
structural elements, while the smaller inner rings, which come into contact
with the PTFE reflectors, facilitate charge removal. The small dimension of
the inner rings is necessary to minimize the inevitable local field distortion
induced by the presence of conductive elements close to the active volume. For
the same reason, despite their wider surface area, the outer rings have a non-
discernible impact on the local drift field.

The XENONnT field cage was constructed to make contact with the exterior of
the PTFE walls wherever feasible. This decision was prompted by the
observation in XENON1T of an inward push in the reconstructed (or observed)
radial position of the events correlated to the $\left(x,y\right)$ geometry of
the TPC XENON:2019ykp . The radial distortion was explained as a charge-up of
the PTFE walls, exhibiting a time dependence. The azimuthal dependence, known
as “bite-structure” showed a stronger inward push around the panels than
around the pillars. This effect was attributed to a smaller accumulation of
charges on the pillars with respect to the panels, possibly due to a more
efficient removal process resulting from the contact of the XENON1T field cage
with the pillars.

The active region of the XENONnT TPC is a prism with a height of
$148.6\text{\,}\mathrm{cm}\text{/}$ and a 24-sided polygonal cross-section,
with $132.8\text{\,}\mathrm{cm}\text{/}$ between opposite sides. Two PMT
arrays and stacks of electrodes limit the TPC at the top and bottom. The
electrode stacks as well as the PMT arrays are supported by 24 PTFE pillars. A
total of 24 $3\text{\,}\mathrm{mm}\text{/}$ thick PTFE “sliding” panels are
mounted in between the pillars and they interlock with 24 PTFE “blocking”
panels directly mounted on the pillars. A total of 355 clipping notches are
incorporated into each of the sliding panels to maintain contact between the
inner rings of the field cage and the insulator. In addition, the sliding
panels feature $0.25\text{\,}\mathrm{mm}\text{/}$ diameter through-holes at
the center of each clipping notch. They serve the purpose of facilitating the
removal of free charges present on the inner side of the wall, as the mobility
of electrons along the PTFE surface is expected to be larger than across the
material bulk.

The 71 inner rings consist of $2\text{\,}\mathrm{mm}\text{/}$ diameter wires,
taken from a single OFHC copper spool. The wire was first stretched around a
mock-up of $133.1\text{\,}\mathrm{cm}\text{/}$ inner diameter, cut to the
right length, and both ends were threaded. During installation the ends were
connected using polyether ether ketone (PEEK) fasteners, allowing the
circumference to be adjusted by a few millimeters.

The outer field cage array consists of 64 rigid copper rings, each having a
24-sided polygonal shape with a $135.5\text{\,}\mathrm{cm}\text{/}$ distance
between opposite sides and a cross section of [product-units=single]15x5 with
a $2.5\text{\,}\mathrm{mm}\text{/}$ rounding radius. The outer rings are
positioned along $z$ between $-7$ and $-145\text{\,}\mathrm{cm}\text{/}$
($z=0$ being the vertical position of the gate) and with half pitch offset
from the inner field cage rings. Each outer ring consists of two halves
connected by four countersunk M3 stainless steel bolts. One half ring is meant
to be fixed in position and it features two additional holes close to the
junction which are used to connect the resistor chains. The other half ring
can be removed for easier access during maintenance.

The geometry of the field cage is mostly constrained by the compact TPC design
and its key role for the mechanical stability of the detector. The minimal
radial distance between inner and outer field cage arrays is
$8.7\text{\,}\mathrm{mm}\text{/}$. The radial position of the inner rings is
determined by the PTFE wall they are clipped into, whereas the outer radius of
the outer rings is limited by the high-voltage feedthrough (HVFT) running to
the cathode along the full length of the TPC. While a larger radius of the
field cage would improve the drift field homogeneity, a smaller distance
between outer rings and the grounded stainless steel sleeve of the HVFT
increases the risk of discharges.

The vertical distribution of the field cage arrays is constrained at the top
and bottom by the position of the gate and cathode frames. A pitch of
$21.6\text{\,}\mathrm{mm}\text{/}$ at liquid xenon temperature was chosen to
facilitate the assembly of the resistor chain within the reduced intra-array
space. Additional inner rings are included in the design at the top and bottom
ends of the field cage: four rings right below the gate and two above the
cathode (compare to Figure 1). These extra elements are installed with half
the normal pitch. They improve the field homogeneity in regions dominated by
edge effects and field leakage coming from the electrodes’ transparency.

The voltage divider of the field cage is entirely realized using
$5\text{\,}\mathrm{G\SIUnitSymbolOhm}\text{/}$ SMD resistors with
$1\text{\,}\mathrm{\char 37\relax}\text{/}$ tolerance by OHMITE resistor .
These resistors were already used in XENON1T and extensively tested against
failures. They are arranged in order to ensure a linear potential drop along
the $z$-axis. The inner and outer set of rings have independent resistor
chains, which are connected at the top and bottom to form the voltage divider
of the field cage. This minimizes the impact on the electric field in case of
a broken resistor, while simplifying its assembly. Two redundant sets of
voltage dividers are implemented on opposite sides of the TPC.

The electrical connection for the outer rings is achieved by clamping the end
of a $0.4\text{\,}\mathrm{mm}\text{/}$ OFHC wire, soldered to the resistor, to
the countersunk holes via an M3 screw. The resistor is then held in place by a
[product-units=single]7x7 PTFE element inserted between two rings. Given the
small wire diameter of the inner array rings, a spring loaded connection was
realized. Dovetail notches on the reflecting panels support counterpart PTFE
pieces on which the resistors are mounted. This joint establishes a spring-
loaded connection using M1.7 set screws in electric contact with both the
resistors and the inner copper rings. The connection was tested for stress due
to temperature changes using liquid nitrogen, proving a good reliability.

All the materials used to machine and assemble the field cage and its resistor
chains were screened and thoroughly cleaned to ensure radiopurity nt_screening
.

##  3 Electric field simulation

The resistor chain and the design of the field cage was optimized based on the
simulation of the electric field of the TPC. The simulations were performed
using COMSOL Multiphysics® v5.4 comsol , in particular the AC/DC module with
finite element method (FEM) analysis. This method involves the discretization
of the geometry into smaller elements, an operation known as “meshing”. The
electrostatic equations are then solved at the vertices, or “nodes”, of each
element and interpolated in between.

Given the great number of simulations needed during the design and
optimization of the field cage, as well as the high computational power
required for a full 3D simulation, a 2D-axisymmetric model of the detector was
implemented. Decreasing the dimension of the problem reduces the number of
nodes needed in order to be able to simulate the full detector. However, this
excludes non-axisymmetrical features, such as the polygonal structure of the
field cage or the wire grid nature of the electrodes. In the 2D-axisymmetric
simulation the TPC is constructed as a cylinder with electrodes made of
concentric wires. While this approximation impacts the electric field close to
the electrodes, the expected effect in the detector’s active volume was
estimated to be marginal by simulating a small-scale TPC using both 3D and
2D-axysimmetric geometry. Comparing the electric field inside the active
volume, a difference lower than $0.5\text{\,}\mathrm{\char 37\relax}\text{/}$
was found. This is notably smaller than the effect of introducing a charge
distribution on the PTFE walls, as discussed in Sect. 4, and it is thus
considered negligible.

The TPC is contained inside a vacuum-insulated double-walled stainless steel
cryostat, which acts as a Faraday cage. This means that the simulation of the
TPC environment can be restricted to the grounded inner vessel. Given the
2D-axisymmetric nature of the simulation, elements that only cover a small
azimuthal angle are excluded from the model. This includes the PTFE pillars,
the HVFT to the cathode, and the resistor chains. These elements were studied
separately with local 3D simulations in order to evaluate their impact on the
drift field and to assess the risk of breakdown. The PMTs are approximated by
a concentric structure in the 2D simulation. The impact of this approximation
on the drift field is expected to be negligible as they are located behind the
screening grids and far from the active volume.

The dimensions of the TPC elements span several orders of magnitude, ranging
from the $216\text{\,}\mathrm{\SIUnitSymbolMicro m}\text{/}$ diameter of the
electrode wires up to the $1.5\text{\,}\mathrm{m}\text{/}$ length of the
reflector panels. For this reason, the mesh size ranges from
$30\text{\,}\mathrm{\SIUnitSymbolMicro m}\text{/}$ around the electrode wires
up to $25\text{\,}\mathrm{mm}\text{/}$ in the center of the LXe volume, where
the electric field is most uniform. The final mesh consists of
$4.8\text{\times}{10}^{6}$ elements and $2.4\text{\times}{10}^{6}$ nodes. When
the field within the active volume is compared to the same geometry simulated
with a coarser mesh, the average difference is within $1\text{\,}\mathrm{\char
37\relax}\text{/}$, being larger close to the electrodes. Hence, we conclude
that the uncertainty from meshing can be ignored.

As discussed in Sect. 2, the field cage geometry was strongly constrained by
mechanical requirements. For this reason, the uniformity of the electric drift
field was optimized by selecting the voltages applied to the field cage. If
the voltage drop is proportional to the vertical separation of two consecutive
field shaping elements, then the voltage gradient is constant and the electric
field is uniform. The voltages applied to the top and bottom of the field cage
should match the effective potentials in those positions, which differ from
the voltages of gate and cathode due to the field leakage effect previously
described. At the top, this matching is done by independently biasing the
topmost inner field cage ring. This freedom in bias voltage represents an
important innovation, as it enables the tuning of the field homogeneity during
operation of the filled detector. This permits adjusting to different
electrode configurations or exploring the effect of the field homogeneity on
the signal, as done in Sect. 4. An additional HVFT would have been necessary
for a similar solution at the bottom of the field cage, considering the
requirement for voltages as low as $-30\text{\,}\mathrm{kV}\text{/}$. Such a
solution was not implemented. Instead, a fixed resistance between the
bottommost inner field cage array element and the cathode was installed. As it
is not possible to change this resistance once the detector is assembled, its
value was optimized considering the possibility that the design cathode
voltage might not be reached. The electric field inside the XENONnT TPC was
simulated using the electrodes’ design potentials of
$-1\text{\,}\mathrm{kV}\text{/}$ at the gate,
$6.5\text{\,}\mathrm{kV}\text{/}$ at the anode and
$-30\text{\,}\mathrm{kV}\text{/}$ at the cathode. Different combinations of
the topmost inner field cage ring voltage $V_{\mathrm{top}}$ and the bottom
resistance $R_{\mathrm{bot}}$ were considered. Fields were simulated for a
voltage $V_{\mathrm{top}}$ between $-1.2$ and
$-0.5\text{\,}\mathrm{kV}\text{/}$, and a resistance $R_{\mathrm{bot}}$
between $5$ and $10\text{\,}\mathrm{G\SIUnitSymbolOhm}\text{/}$.

Two independent figures of merit were used in order to evaluate the
performance of the different configurations: the field spread within the
$4\text{\,}\mathrm{t}\text{/}$ fiducial volume (FV) as defined in nt_mc , and
the size of the charge-insensitive volume (CIV). The field spread is defined
as the difference between the 5 and 95 percentile of the electric field
magnitude divided by its mean. The charge-insensitive volume is a region of
the detector characterized by the complete or partial loss of the ionization
electrons. The electrons freed in such a volume follow the electric field
lines ending on the PTFE walls, accumulating on the wall and thus not
producing S2 signals. The reverse field region between the cathode and the
bottom PMT array is an example of irreducible CIV, and it is therefore ignored
in this discussion. The CIV is calculated by propagating the electrons along
the simulated field lines from different positions within the TPC active
volume and checking whether they end up on the wall surface or reach the
liquid-gas interface. These figures of merit were computed using the custom
module PyCOMes pycomes , developed to handle COMSOL output format and perform
fast calculations of field lines and electron propagation. The mass of liquid
xenon inside the CIV ($M_{\mathrm{CIV}}$) is shown in the ($V_{\mathrm{top}}$,
$R_{\mathrm{bot}}$) parameter space in Fig. 3.

Figure 3: Charge-insensitive mass $M_{\mathrm{CIV}}$ as a function of the
voltage of the topmost inner field cage ring $V_{\mathrm{top}}$ and the
resistance $R_{\mathrm{bot}}$ between the bottom of the field cage and the
cathode. The contour lines represent the relative drift field spread within
the $4\text{\,}\mathrm{t}\text{/}$ fiducial volume. The red star shows the
configuration picked for XENONnT, with
$V_{\mathrm{top}}$$=$$-0.95\text{\,}\mathrm{kV}\text{/}$ and
$R_{\mathrm{bot}}$$=$$7\text{\,}\mathrm{G\SIUnitSymbolOhm}\text{/}$.

When the topmost inner field cage ring is biased more positively than the gate
electrode, the electrons drifting through the TPC are more strongly attracted
to it. This improves the uniformity of the drift field within a limited range
of voltages. As the bias voltage increases further relative to the gate
electrode, the field lines begin to terminate on the PTFE wall, and at
approximately $V_{\mathrm{top}}=-0.85\,\mathrm{kV}$, the CIV abruptly
increases. A larger CIV is also observed for high values of
$R_{\mathrm{bot}}$. This is due to an increasing local field distortion in the
bottom edge of the detector.

Compromising between field homogeneity and CIV, the values
$V_{\mathrm{top}}=-0.95\,\mathrm{kV}$ and
$R_{\mathrm{bot}}=7\,\mathrm{G\Omega}$ were chosen for the XENONnT design
field. This corresponds to $M_{\mathrm{CIV}}=1.2\,$\mathrm{kg}\text{/}$$ and a
field spread of $3.5\text{\,}\mathrm{\char 37\relax}\text{/}$. We checked the
performance of the electric field with the bottom resistance value
$R_{\mathrm{bot}}$ for different configurations of the electrode voltages,
with the result that the chosen resistance performs sufficiently well for a
wide range of scenarios.

##  4 Comparison to data

The XENONnT detector is periodically calibrated using ^83mKr. The metastable
isotope has a half-life of $1.83\text{\,}\mathrm{h}\text{/}$ and decays via a
two-step transition of $32.2\text{\,}\mathrm{keV}\text{/}$ and
$9.4\text{\,}\mathrm{keV}\text{/}$ with an intervening half-life of
$157\text{\,}\mathrm{ns}\text{/}$ kr83m . This source is used to monitor the
spatial response of the detector and its time evolution, assuming its
homogeneous distribution kr_lux ; XENON:2019ykp . It is therefore possible to
compare the observed ^83mKr event distribution to the expected one from
simulations. The simulated distribution comes from a set of electrons
uniformly produced within the active volume. Each electron is propagated
according to the electric field map including diffusion and drift values as
coming from literature exo_diffusion ; nikhef_diffusion . The
$\left(x,y\right)$-position is the electron location at the liquid-gas
interface including the position reconstruction resolution, while the $z$
information is derived from the drift time.

###  4.1 SR0 field and wall charge-up matching

During the commissioning phase of the experiment, a short circuit occurred
between the cathode and the bottom screen electrode, limiting the voltage of
the cathode. For the first science run (SR0), the electrodes were set to a
voltage of $0.3\text{\,}\mathrm{kV}\text{/}$ at the gate,
$4.9\text{\,}\mathrm{kV}\text{/}$ at the anode and
$-2.75\text{\,}\mathrm{kV}\text{/}$ at the cathode. This resulted in an
average electric drift field of $23\text{\,}\mathrm{V}\text{/}\mathrm{cm}$.
The topmost inner field cage ring voltage was set to
$V_{\mathrm{top}}=0.65\,\mathrm{kV}$, which was optimized based on
simulations, by means of the procedure described in Sect. 3.

Similarly to XENON1T, an azimuthally dependent distortion at high radii is
observed in the ^83mKr distribution, reflecting the 24-sided-polygonal
structure of the PTFE walls. Nevertheless, the distortion shows a different
behaviour than what was observed by XENON1T. In XENONnT the events around the
pillars are pushed more inwards than the events around the panels, leading to
a localized strong reduction of the rate, as shown in Fig. 4.

Figure 4: Reconstructed $\left(x,y\right)$-position distribution of ^83mKr
events. The distortion at high radii follows the distribution of PTFE pillars
and panels, the cross section of which are overlaid in the figure. The
diagonal features crossing the TPC result from the transverse wires of the
gate electrode and the distribution of the PMTs in the top array.

The observed “bite-structure” supports the XENON1T hypothesis of PTFE charge-
up discussed in Sect. 2, which drove the field cage design of XENONnT. While
the field cage rings touch both panels and pillars, the panels are expected to
release accumulated charges more easily than pillars because of their thinner
geometry and the presence of through-holes. The efficient charge removal is
supported by the absence of time-dependent features in the reconstructed
radial position of the ^83mKr events. Figure 5 shows the evolution over SR0 of
the 90 percentile of the radius distribution of ^83mKr events in three slices
of $z$.

Figure 5: Evolution of the 90 percentile of the radial distribution at three
different $z$ bins centered around the indicated value and with a bin width of
$7\text{\,}\mathrm{cm}\text{/}$.

Unlike previous experiments lux_3d_field ; XENON:2019ykp , no increase of the
inward push is observed over time. In addition, the observed
$\left(x,y\right)$-distribution has two symmetric features crossing the TPC as
a result of the transverse wires of the gate locally deflecting the electrons
when drifting towards the liquid-gas interface. These wires were installed
both at the gate and anode electrodes to counteract wire deformation under
electrostatic force nt_instrument_paper . The regular pattern perpendicular to
these features is due to a combination of the wire grid of the anode electrode
with the geometrical configuration of the PMTs in the top array.

The 90 percentile of the radial distribution $r_{90}$ is evaluated for both
^83mKr data and simulation in 30 bins of $z$ by averaging over the azimuthal
angle. The simulation consists in the propagation of $10^{5}$ electrons with
initial position uniformly distributed in the active volume. The 90 percentile
is sufficiently large not to be affected by the transverse wires, but not
sensitive to possible outliers at high percentiles. A mismatch between data
and simulation can be clearly seen in Fig. 6, with the difference between
$r_{90}$ from data (black circles) and from simulation (blue triangles) being
on average $4.7\text{\,}\mathrm{cm}\text{/}$. The mismatch can be effectively
resolved by considering a charge accumulation on the PTFE walls, as already
demonstrated in previous works lux_3d_field . The corresponding surface charge
density $\sigma_{\mathrm{w}}$ is determined by matching the observed radial
distribution with simulations including a charge distribution at the walls.
This density is parameterized using a linear function:

$\sigma_{\mathrm{w}}=\lambda\cdot\frac{\left|z\right|}{h_{\mathrm{TPC}}}+\sigma_{\mathrm{top}},$
(1)

where $h_{\mathrm{TPC}}=148.6\,\mathrm{cm}$ is the height of the TPC,
$\sigma_{\mathrm{top}}$ is the surface charge density at the top of the panels
and $\lambda$ is the charge density difference between the top and bottom of
the panels, i.e., $\sigma_{\mathrm{bot}}=\sigma_{\mathrm{top}}+\lambda$.

Figure 6: ($r$,$z$) distribution of ^83mKr events near the walls of the TPC.
The 90 percentile of the radial distribution along 30 bins of $z$ is shown in
black. The same quantities coming from the simulation with and without PTFE
reflector charge-up are shown in blue and orange, respectively.

A linear model describes to first order the observations reported in
lux_3d_field . Field and electron-propagation simulations were performed for
$\sigma_{\mathrm{top}}$ between $-2$ and
$1.5\text{\,}\mathrm{\SIUnitSymbolMicro C}\text{/}{\mathrm{m}}^{2}$ and for
$\lambda$ between $-1$ and $1.5\text{\,}\mathrm{\SIUnitSymbolMicro
C}\text{/}{\mathrm{m}}^{2}$, both with steps of
$0.1\text{\,}\mathrm{\SIUnitSymbolMicro C}\text{/}{\mathrm{m}}^{2}$. For each
combination of $\sigma_{\mathrm{top}}$ and $\lambda$, $r_{90}^{\mathrm{sim}}$
was calculated for $n_{z}=30$ bins in $z$ along the TPC as described above.
The chi-square was estimated for each simulation in the following way:

$\chi^{2}=\sum_{i=0}^{n_{z}-1}\frac{\left(r_{90,i}^{\mathrm{obs}}-r_{90,i}^{\mathrm{sim}}\right)^{2}}{\sigma_{90,i}^{2}},$
(2)

where $\sigma_{90}^{2}$ is the squared sum of the statistical percentile
uncertainties of data and simulations. The $\chi^{2}$ best fit yields

$\displaystyle\sigma_{\mathrm{top}}$
$\displaystyle=\left($-0.50(0.06)$\text{\scriptsize{(syst)}}\pm
0.02\text{\scriptsize{(stat)}}\right)\,$\mathrm{\SIUnitSymbolMicro
C}\text{/}{\mathrm{m}}^{2}$,$ $\displaystyle\lambda$
$\displaystyle=\left($0.40(0.15)$\text{\scriptsize{(syst)}}^{+$0.20$}_{$-0.10$}\text{\scriptsize{(stat)}}\right)\,$\mathrm{\SIUnitSymbolMicro
C}\text{/}{\mathrm{m}}^{2}$.$

These values correspond to a surface charge density of
$-0.5\text{\,}\mathrm{\SIUnitSymbolMicro C}\text{/}{\mathrm{m}}^{2}$ at the
top of the panels and $-0.1\text{\,}\mathrm{\SIUnitSymbolMicro
C}\text{/}{\mathrm{m}}^{2}$ at the bottom. The statistical uncertainty was
determined by resampling the simulated position distributions for each
parameter combination, a technique known as “bootstrapping”, and then
assessing their $\chi^{2}$ best fit. The systematic uncertainty was obtained
by repeating the $\chi^{2}$ minimization with different binning in $z$ and
percentile values, and taking into account the coarse binning for the
$\sigma_{\mathrm{top}}$ and $\lambda$ parameters.

The simulated radial distribution after adding the wall charge-up component is
shown as cyan squares in Fig. 6, with a maximum difference with respect to the
observed distribution of $1.6\text{\,}\mathrm{cm}\text{/}$ at the very top of
the TPC and $0.3\text{\,}\mathrm{cm}\text{/}$ on average. Including the
surface charge density, the predicted field spread is
$13.2\text{\,}\mathrm{\char 37\relax}\text{/}$ within the FV and the charge-
insensitive mass is $112\text{\,}\mathrm{kg}\text{/}$. The corresponding SR0
electric field map including charge-up is shown in Fig. 7.

Figure 7: Electric field map determined from 2D-axisymmetric simulations
including a linear charge distribution on the PTFE reflectors matched to the
radial distribution of ^83mKr events. The black lines indicate the contour of
the electric field, while the dashed grey lines are field lines starting at
different radii and same $z$.

The hypothesis that a failure of the resistor chain causes the mismatch
between the simulation and the data can be ruled out, as the total resistance
of the field cage was measured to be $92\pm
11.\text{\,}\mathrm{G\SIUnitSymbolOhm}\text{/}$, in good agreement with the
expected value of $87.25\pm 0.05\text{\,}\mathrm{G\SIUnitSymbolOhm}\text{/}$.
Moreover, the simulation of the failure of a single resistor shows an
insufficient impact on the observed position distribution.

An independent validation of the field map including wall charge-up comes from
the measurement of the electron lifetime during SR0. The electron lifetime
$\tau_{e^{-}}$ is the characteristic time constant of the exponential decrease
of the S2 signal as a function of drift time $t_{d}$. This is due to electrons
being trapped by impurities in the liquid xenon. To determine the electron
lifetime $\tau_{e^{-}}$, an exponential function is fitted to the median of
the S2 area across different drift times. Previous analyses of XENON1T data
revealed a discrepancy in the measurement of the electron lifetime when using
different radioactive isotopes, such as ^83mKr and ^222Rn XENON:2019ykp .
These isotopes vary in their decay products and energy, resulting in different
ionization densities within xenon. For this reason, the electric field affects
the charge signal of each calibration source differently, leading to a
different spatial dependence of the S2 signal in presence of an inhomogeneous
drift field. The electron lifetime measured according to the above approach is
thus an effective value $\tau_{e^{-}}^{\mathrm{eff}}$ that includes a relative
field effect on the charge yield,
$\mathit{Q_{\mathrm{y}}^{\mathrm{rel}}}(x,y,z)$:

Figure 8: Electron lifetime measured using three different radioactive
isotopes: ^37Ar (light blue squares), ^83mKr (blue triangles) and ^222Rn
(black circles). The results on the left plot use the uncorrected charge
signals, while middle and right plots include a drift field correction based
on the field map without and with charge on the PTFE wall, respectively. The
data were taken during a simultaneous ^37Ar and ^83mKr calibration after the
end of SR0. Due to emanation, minute levels of ^222Rn are present in the
detector at all times nt_screening .

$\displaystyle\mathrm{S2}\left(t_{d}\right)$
$\displaystyle=\mathrm{S2}\left(0\right)\cdot\exp{\left(-t_{d}/\tau_{e^{-}}^{\mathrm{eff}}\right)}$
(3)
$\displaystyle=\mathrm{S2}\left(0\right)\cdot\mathit{Q_{\mathrm{y}}^{\mathrm{rel}}}\left(x,y,z\right)\cdot\exp{\left(-t_{d}/\tau_{e^{-}}\right)}.$

Figure 8 shows the results from the measurement of the electron lifetime
during a joint calibration using ^37Ar and ^83mKr sources, which was carried
out after the end of SR0 lower_nt . The “uncorrected” electron lifetime comes
from the exponential fit of the S2 median area, and it corresponds to
$\tau_{e^{-}}^{\mathrm{eff}}$ of Eq. (3). The “corrected” lifetime is obtained
by weighting the measured S2 area for the relative charge yield
$\mathit{Q_{\mathrm{y}}^{\mathrm{rel}}}$ as coming from the field map. This is
the best estimate of the true electron lifetime $\tau_{e^{-}}$. Since each
calibration source is affected differently by the electric field,
$\mathit{Q_{\mathrm{y}}^{\mathrm{rel}}}$ is estimated for each isotope. The
^83mKr charge yield is modeled using data from heidelberg_hexe . ^37Ar data
are extrapolated to lower electric fields using the results from pixey . The
charge yield of ^222Rn alphas finally is modeled using NEST v1 nest_paper ;
nestv1 , which is more consistent with recent measurements than the latest
version heidelberg_hexe . When not corrected, the electron lifetimes from
different sources (left panel) do not agree among themselves, as this
corresponds to assuming a perfectly homogeneous electric drift field. The
lifetimes are corrected using the simulated electric field maps both with
(right panel) and without (middle panel) the inclusion of the charge-up of the
PTFE walls. The different measurements are in agreement when charge
accumulation is assumed, but when it is not included, the discrepancy is even
more pronounced than in the uncorrected case.

###  4.2 Impact of the field cage tuning on the drift field

Figure 9: Reconstructed position distribution of ^83mKr events for different
voltages of the topmost inner ring of the field cage. Red circles and black
triangles are the 90 percentile radial distribution coming from simulation and
data, respectively. The TPC active volume boundaries are shown as black dashed
lines.

Dedicated datasets were taken after the end of SR0 to assess the impact of
different voltages of the topmost inner field cage ring $V_{\mathrm{top}}$ on
the drift field. This was varied from $0.3\text{\,}\mathrm{kV}\text{/}$ to
$1\text{\,}\mathrm{kV}\text{/}$ during a ^83mKr calibration, while keeping the
voltages of all other electrodes at their SR0 values. While the electrode
voltages mostly impact the magnitude of the electric field, the independent
biasing of the topmost inner field cage ring influences primarily its
homogeneity. From simulations, changing $V_{\mathrm{top}}$ from
$0.3\text{\,}\mathrm{kV}\text{/}$ (same as gate) to
$1\text{\,}\mathrm{kV}\text{/}$ translates into a $7\text{\,}\mathrm{\char
37\relax}\text{/}$ stronger field within the FV, while reducing the field
spread by $400\text{\,}\mathrm{\char 37\relax}\text{/}$. For this reason, this
is the first direct measurement of the effect of the field homogeneity on
signal production and the transport of S2 electrons in a multi-tonne LXe TPC.

The reconstructed ^83mKr $\left(r,z\right)$ distribution and the 90 percentile
radial distribution are shown in Fig. 9 for different field cage tuning
voltages. As $V_{\mathrm{top}}$ increases, a decreasing radial inward push is
observed. As discussed in Sect. 3, a more positive voltage at the top of the
TPC attracts electrons counteracting the inward push, resulting in a more
uniform distribution. However, by increasing $V_{\mathrm{top}}$ the charge-
insensitive volume increases. The CIV cannot be inferred from the observed
position distribution even for $V_{\mathrm{top}}>0.75\,\mathrm{kV}$, when
$>10\,$\mathrm{\char 37\relax}\text{/}$$ of the total TPC volume is charge-
insensitive. For these configurations, the edge of the position distribution
is flat over $z$, showing no inward feature.

The comparison of $r_{90}^{\mathrm{obs}}$ for different $V_{\mathrm{top}}$
with the corresponding $r_{90}^{\mathrm{sim}}$ including the SR0 wall charge
distribution returns a good match for voltages $V_{\mathrm{top}}$below or
equal to $0.75\text{\,}\mathrm{kV}\text{/}$. A difference of up to
$5\text{\,}\mathrm{cm}\text{/}$ in the 90 percentile radial distribution is
observed for a $V_{\mathrm{top}}$ of $0.9\text{\,}\mathrm{kV}\text{/}$ and
$1\text{\,}\mathrm{kV}\text{/}$. This hints towards a mismodeling of the
charge distribution or the possibility that the charge distribution reaches a
new equilibrium for high voltages $V_{\mathrm{top}}$.

Figure 10: Rate of events (light blue squares), signals S1 (black circles) and
S2 signals (blue triangles) for the same ^83mKr source, but different topmost
inner field cage ring voltages. All rates are corrected for the ^83Rb source
decay.

The change of the CIV is reflected by the change of the observed event and
signal rates for different $V_{\mathrm{top}}$ values, as shown in Fig. 10. An
event is defined by the pairing of an S1 and S2 signal lower_nt  passing a
loose box ^83mKr selection on their area, while the individual signals (S1 or
S2) are selected within the corresponding range, but without pairing
requirement. The time elapsed between the beginning and the end of the test is
around one day. Because of the ^83Rb half-life of
$86.2\text{\,}\mathrm{d}\text{/}$, a daily decrease in rate of $1.01\pm
0.12\text{\,}\mathrm{\char 37\relax}\text{/}$ is considered in the
calculation, verified by comparing the rate before and after the test using
the same SR0 field configuration. As expected, the S1 rate is constant for
different $V_{\mathrm{top}}$, while S2 and event rates are fairly constant up
to $0.75\text{\,}\mathrm{kV}\text{/}$, but quickly drop for larger values.
This observations proves that a fraction of the active volume is charge-
insensitive and that this depends on the electric field configuration. The
fast increase of $M_{\mathrm{CIV}}$ is explained by the anodic behaviour of
the topmost inner field cage ring. At these voltages the drifting electrons
are collected on the PTFE walls at the very top of the TPC. In this situation
even a small change of $V_{\mathrm{top}}$ leads to a large fraction of field
lines being lost at the edges, although the impact on the intensity of the
field is negligible.

Thanks to the large rate of ^83mKr events collected during the test, it is
possible to measure the electron lifetime individually for each voltage. A
clear dependence on $V_{\mathrm{top}}$ is shown in Fig. 11. The observed
increase of the uncorrected electron lifetime (black circles) is explained by
the lower electric field at the top of the TPC as $V_{\mathrm{top}}$
increments, as it is suggested by the field simulations. A smaller electric
field leads to a reduced charge yield, finally resulting in an higher
uncorrected electron lifetime. As these data have been taken within few hours,
the fast change of the electron lifetime as $V_{\mathrm{top}}$ increases
cannot be due to a change of the impurity concentration in the liquid xenon.
Similarly, the small variation of the electric drift field for different
$V_{\mathrm{top}}$ values cannot account for the change of an order of
magnitude in the electron lifetime attachment . Finally, the uncorrected
electron lifetime measured right before and after the test in standard field
conditions agrees well, excluding a possible evolution over time of the
lifetime.

The effects due to the non-uniform electric field on the uncorrected electron
lifetime can be accounted for by simulating the electric field, similarly to
what is done in Fig. 8. Figure 11 shows the corrected electron lifetime for
different values of $V_{\mathrm{top}}$. The corrected values agree with the
constant average of $38\text{\,}\mathrm{ms}\text{/}$, considered to be the
true electron lifetime $\tau_{e^{-}}$ and further demonstrating the capability
to correct for the electric field effect solely based on simulations.

Figure 11: Electron lifetime measured using ^83mKr for different topmost inner
field cage ring voltages. Blue triangles and black circles are electron
lifetime with and without field correction, respectively. The cyan dashed line
corresponds to $38\text{\,}\mathrm{ms}\text{/}$, which is the average value of
measurements with field correction.

##  5 Summary

This work demonstrated a good understanding and effective control of the
electric field inside the active volume of the XENONnT TPC. The novel double-
array structure of the field cage allows for mechanical stability, while
ensuring contact between the conducting field shaping elements and the PTFE
walls, facilitating the removal of charges accumulating over time. The absence
of a time evolution in the distribution of the observed event position
confirmed an efficient removal.

The innovative independent voltage bias of the topmost field cage ring makes
it possible to match it to the local effective potential, a combination
between gate and anode voltages due to field leakage through the gate. The
detector was simulated using the FEM software COMSOL Multiphysics®, using an
approximate 2D-axisymmetric geometry. The bias voltage of the topmost inner
field cage ring and the value of the resistor between the field cage and the
cathode were chosen to optimize the charge-insensitive volume and the field
homogeneity.

During SR0, the spatial distribution of ^83mKr calibration data was compared
to the one calculated based on the electric field simulation. A linear surface
charge density along the PTFE walls of the TPC was included in the field
simulation to improve the agreement of the reconstructed position distribution
between simulation and data. The best match to data was obtained with a charge
density distribution ranging from $-0.5\text{\,}\mathrm{\SIUnitSymbolMicro
C}\text{/}{\mathrm{m}}^{2}$ at the top of the walls to
$-0.1\text{\,}\mathrm{\SIUnitSymbolMicro C}\text{/}{\mathrm{m}}^{2}$ at the
bottom, reducing the average difference between simulated and observed 90
percentile radial distribution from $4.7\text{\,}\mathrm{cm}\text{/}$ down to
$0.3\text{\,}\mathrm{cm}\text{/}$. The resulting field map was used to correct
the relative charge yield of S2 signals used for the estimation of the
electron lifetime from different sources. This resolved a long-standing
discrepancy and further validated the simulations.

A dedicated test to investigate the impact of the topmost inner field cage
ring voltage on the field uniformity was performed using the data from a
^83mKr calibration source. An average difference
$<1\text{\,}\mathrm{cm}\text{/}$ of the 90 percentile radial distribution is
observed between data and simulations when including the reflector charge-up
for voltages below $0.75\text{\,}\mathrm{kV}\text{/}$. Above this value, a
deteriorating agreement in the position distribution, together with a strong
decrease in event and S2 rates, indicating a significant increase of charge-
insensitive volume. The measured electron lifetime as a function of the
topmost inner field cage ring voltage showed an apparent increase of an order
of magnitude, which cannot be explained by the change of impurity
concentrations. However, as the S2 signals are corrected for the field
dependent charge yields evaluated using the proper electric field map, the
electron lifetime measurements for the different runs agree within the
uncertainties.

The presented design of the field cage for the XENONnT TPC represents a
novelty for the dual-phase TPC technology, allowing for control over the
homogeneity of the field while minimizing known effects of charge accumulation
on the detector walls. Together with the good understanding of the electric
drift field, this elevates the capability of TPC detectors for dark matter
searches improving the sensitivity to WIMPs and potentially setting a new
standard in the field.

###### Acknowledgements.

We gratefully acknowledge support from the National Science Foundation, Swiss
National Science Foundation, German Ministry for Education and Research, Max
Planck Gesellschaft, Deutsche Forschungsgemeinschaft, Helmholtz Association,
Dutch Research Council (NWO), Weizmann Institute of Science, Israeli Science
Foundation, Binational Science Foundation, Région des Pays de la Loire, Knut
and Alice Wallenberg Foundation, Kavli Foundation, JSPS Kakenhi and JST FOREST
Program in Japan, Tsinghua University Initiative Scientific Research Program
and Istituto Nazionale di Fisica Nucleare. This project has received
funding/support from the European Union’s Horizon 2020 research and innovation
programme under the Marie Skłodowska-Curie grant agreement No 860881-HIDDeN.
Data processing is performed using infrastructures from the Open Science Grid,
the European Grid Initiative and the Dutch national e-infrastructure with the
support of SURF Cooperative. We are grateful to Laboratori Nazionali del Gran
Sasso for hosting and supporting the XENON project.

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