# The upgraded low-background germanium counting facility Gator for high-
sensitivity $\gamma$-ray spectrometry

Gabriela R. Araujo  Laura Baudis  Yanina Biondi  Alexander Bismark  Michelle
Galloway

###### Abstract

We describe the upgrade and performance of the high-purity germanium counting
facility Gator, which is dedicated to low-background $\gamma$-ray
spectrometry. Gator is operated at the Gran Sasso Underground Laboratory in
Italy, at an average depth of 3600 meter water equivalent, and employed for
material screening and selection in ultra-low background, rare-event search
experiments in astroparticle physics. The detector is equipped with a passive
shield made of layers of copper, lead and polyethylene, and the sample cavity
is purged with gaseous nitrogen maintained at positive pressure for radon
suppression. After upgrading its enclosure, the background rate is $82.0\pm
0.7\text{\,}\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{n}\mathrm{t}\mathrm{s}\mathrm{/}\mathrm{(}\mathrm{k}\mathrm{g}\cdot\mathrm{d}\mathrm{a}\mathrm{y}\mathrm{)}$
in the energy region
$100\text{\,}\mathrm{k}\mathrm{e}\mathrm{V}2700\text{\,}\mathrm{k}\mathrm{e}\mathrm{V}$,
a $20\text{\,}\mathrm{\char 37\relax}$ reduction compared to the previously
reported rate. We show the stability of various operation parameters as a
function of time. We also summarize the sample analysis procedure, and
demonstrate Gator’s sensitivity by examining one material sample, a candidate
photosensor for the DARWIN experiment.

##  1 Introduction

Gamma-ray spectrometers based on ultra-low background, high-purity germanium
(HPGe) detectors are routinely employed to screen and select detector
materials for rare-event searches [1]. Germanium spectrometry offers a non-
destructive screening method and provides information on material
radioactivity levels in a single energy spectrum with excellent resolution for
$\gamma$-ray lines emitted by primordial (238U, 232Th, 40K), cosmogenic (e.g.
54Mn, 46Sc, 60Co), and anthropogenic (e.g. 137Cs, 110mAg) isotopes.
Sensitivities for HPGe detectors located in underground laboratories are at
the level of
$10-$100\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{B}\mathrm{q}\mathrm{/}\mathrm{k}\mathrm{g}$$
for large masses (tens of kg) and long counting times (1-3 months) [2, 3, 4,
5, 6, 7].

The Gator facility, located at the Laboratori Nazionali del Gran Sasso (LNGS)
of INFN in Italy and described in detail in [3], has been employed to select
materials for the dark matter search experiments XENON100 [8], XENON1T [9] and
XENONnT [10], as well as for the neutrinoless double beta decay experiments
GERDA [11] and LEGEND-200. Radioassay of potential materials for the DARWIN
[12] experiment is ongoing and will also be performed for LEGEND-1000 [13] in
the near future. We have upgraded the facility in order to decrease the
overall background level as well as to facilitate the sample handling process.
The aim of this paper is to describe the facility modifications and their
impact on the background rate with respect to [3] as well as provide an update
on the detector operation and calibration. We also present the sample analysis
procedure, illustrated with a sample from the DARWIN material screening
campaign. This article is organised as follows: In section 2 we describe the
improvements to the system in detail. In section 3 we describe the detector
operation and general performance, and we discuss the new background level in
section 4. In section 5 we present the analysis procedure along with results
for a DARWIN sample which was measured after the upgrade. We provide a summary
and outlook in section 6.

##  2 Description of the detector and its upgrade

Gator deploys a p-type coaxial HPGe detector with a
$2.2\text{\,}\mathrm{k}\mathrm{g}$ sensitive mass and a relative efficiency of
$100.5\text{\,}\mathrm{\char 37\relax}$111Detection efficiency given for the
$1.33\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$ $\gamma$-ray from 60Co and
defined relative to a 3-inch × 3-inch NaI(Tl) crystal placed at a distance of
$25\text{\,}\mathrm{c}\mathrm{m}$ from the source [14]. [3]. The HPGe crystal
is placed in an ultra-low background, oxygen-free copper cryostat and
surrounded by several layers of shielding material. The innermost
$5\text{\,}\mathrm{c}\mathrm{m}7\text{\,}\mathrm{c}\mathrm{m}$-thick shield is
made of oxygen-free high-conductivity copper, which is surrounded by 5-cm
inner and 15-cm thick outer layers of lead with activities of
$3\text{\,}\mathrm{B}\mathrm{q}\mathrm{/}\mathrm{k}\mathrm{g}$ and
$75\text{\,}\mathrm{B}\mathrm{q}\mathrm{/}\mathrm{k}\mathrm{g}$, respectively.
The outermost, 5-cm thick shield is made of borated polyethylene in order to
reduce the ambient neutron flux. The sample cavity has inner dimensions of
25$\times$25$\times$$33\text{\,}\mathrm{c}\mathrm{m}^{3}$ and is continuously
purged with gaseous nitrogen ($\mathrm{GN}_{2}$) at positive pressure to
displace environmental radon. The detector is operated at cryogenic
temperatures, with cooling provided by a copper coldfinger immersed in liquid
nitrogen ($\mathrm{LN}_{2}$). The detector design and operation follow all the
relevant recommendations of ISO 20042 on technical requirements for
$\gamma$-spectrometry methods [15]. For details, we refer to [3].

During the first ten-year period of operation at LNGS (2007-2017), the
shielding structure was surrounded by an aluminum housing with an acrylic
glove box placed above it for sample handling, as shown in [3]. An airlock
system attached to the glove box allowed for pre-purging with
$\mathrm{GN}_{2}$ as well as storage of samples within a nitrogen atmosphere
prior to insertion into the sample cavity. The latter allowed for the decay of
222Rn
($\mathrm{T}_{1/2}=$3.82\text{\,}\mathrm{d}\mathrm{a}\mathrm{y}\mathrm{s}$$),
which may be introduced during sample exchange. In this initial design, both
the geometry of the airlock and the acrylic-to-aluminum interface of the glove
box made it difficult to optimally purge against radon, as well as to access
the cavity for sample loading. Motivated by these problems and by the
requirement to clean the cavity and shield after years of operation, the
enclosure and airlock were redesigned, fabricated, and installed.

Figure 1: Section view of the Gator setup after the upgrade. The HPGe crystal
is housed inside a copper cryostat (1) within the sample cavity, which is
purged with ($\mathrm{GN}_{2}$). The cavity is formed by surrounding layers of
low-background copper (2), lead (3) and polyethylene (4). These layers were
disassembled, cleaned and rebuilt within the new enclosure (5). The load-lock
chamber (6) allows for pre-purging of samples. The acrylic plate (7) and glove
ports (8) for sample viewing and handling are also shown.

The new enclosure, made of stainless steel, is shown schematically in Figure
1. The base of the enclosure consists of two vertically stacked, rectangular
pieces and a flat top plate, with each layer sealed by a continuous rubber
gasket. In place of the acrylic glove box is a flat acrylic plate mounted onto
the top plate and sealed by a rubber gasket, which houses several glove ports.
These design aspects significantly improved the hermeticity of the system,
thus prohibiting the entry of radon via air leaks. The acrylic plate
accommodates several glove ports that allow for top-loading of samples,
facilitating access to the entire volume of the cavity. A removable flange
housing the electrical connections is also installed onto the top plate and
sealed with a rubber gasket. This flange additionally contains ports that
allow for regular nitrogen fills, as the ($\mathrm{LN}_{2}$) dewar connected
to the detector coldfinger is within the enclosure.

The acrylic airlock was replaced by a stainless steel, circular load-lock
chamber (shown to the right of the enclosure, Figure 1), where samples are
pre-purged with $\mathrm{GN}_{2}$ before being placed inside the cavity. Once
purged, the sliding shield on the top of the cavity is opened, followed by the
gate valve that separates the two volumes. The sample is then directly loaded
into the cavity from the inside of the load-lock chamber.

To prepare for installation, the old enclosure was removed and the lead and
copper shielding disassembled. The innermost shielding layers were cleaned in
an ultrasonic ethanol bath in order to remove 40K, oils, and particulates from
surfaces. The structure was then rebuilt within the base of the new enclosure.
Afterwards the $\mathrm{GN}_{2}$ flow was optimized for the new configuration.
The improved hermiticity allowed for a 2/3 reduction in flow at the inlet
while maintaining the same outlet flow as in the previous enclosure. Further
optimisations of the system include the implementation of hermetic seals on
all of the electrical connections that pass through the enclosure and
additional shielding on detector cabling to reduce noise. The integration of a
normally-closed switch on the power line was also added to prevent the
automatic ramp-up of the high voltage module following a power outage.

##  3 Detector operation and performance

The HPGe detector is operated at full depletion under a bias voltage of
$4.8\text{\,}\mathrm{k}\mathrm{V}$. The applied voltage, along with other
parameters that are critical indicators of detector performance and possible
failure (leakage current and $\mathrm{LN}_{2}$ level) are monitored: Their
values are written into a database every two minutes. If exceeding a specified
range, alarms are triggered and sent via email and SMS. We also monitor the
room temperature, which could affect the gain of the amplifier and thus the
calibration of the detector [15].

The stability of the high voltage, the leakage current, $\mathrm{LN}_{2}$
refill cycles and $\mathrm{GN}_{2}$ flow in a period of 2 months in late 2021
are shown in Figure 2 as an example. The values of the monitored parameters
can also be used to correlate operations on the detector or environmental
changes with artefacts in the data. For example, we found an increase in low-
energetic noise, up to $150\text{\,}\mathrm{k}\mathrm{e}\mathrm{V}$, during
refills of the $\mathrm{LN}_{2}$ dewar. This information is employed for an
unbiased removal of these periods from the data, leading to a live-time
reduction of about $2\text{\,}\mathrm{\char 37\relax}$.

Figure 2: Time evolution of selected Gator operational parameters: the
$\mathrm{LN}_{2}$ level in the dewar, the $\mathrm{GN}_{2}$ purge flow, as
well as the detector bias voltage and leakage current. Data points, shown in
6-hour bins, are from a 2-month period towards the end of 2021. In the top
left plot, the weekly refill cycles of the $\mathrm{LN}_{2}$ dewar, which is
connected to the coldfinger of the detector, are clearly discernible as a
sharp rise during each refill, followed by a continuous decrease in level as
the $\mathrm{LN}_{2}$ evaporates. The fluctuations of the $\mathrm{GN}_{2}$
purge flow, detector bias voltage and leakage current are at the sub-percent
level and do not affect the energy calibration.

Regular calibrations of the detector with radioactive sources such as 228Th,
137Cs, or 60Co, as well as with certified extended sources and composite,
high-activity materials, ensure proper knowledge of the detected energy and
its resolution. For this purpose, $\gamma$-ray peaks of selected isotopes are
modeled with a combination of a Crystal Ball function (a Gaussian core part
for the peak with a low-end power-law tail to model loss effects [16]), a
linear background for the continuum, and a smeared step function for
background discontinuities at Compton edges. An exemplary 228Th calibration
spectrum is shown in Figure 3.

Figure 3: A 228Th spectrum calibrated with the known positions of its peaks.
The inset shows a fit of the $860.5\text{\,}\mathrm{k}\mathrm{e}\mathrm{V}$
208Tl line with a combination of a Crystal Ball function, a linear background
as well as a smeared step function.

The determined peak positions are fitted to their literature values with a
polynomial of order one. The energy-dependent resolution is described by a
quadratic function for the variance, resulting in a FWHM at
$1332\text{\,}\mathrm{k}\mathrm{e}\mathrm{V}$ of $2.03\pm
0.04\text{\,}\mathrm{k}\mathrm{e}\mathrm{V}$. This value is shown in Table 1,
along with the energy resolution of some of the most sensitive HPGe detectors
for $\gamma$-ray spectrometry.

Table 1: Location, mass, efficiency, energy resolution (FWHM at $1332\text{\,}\mathrm{keV}$) and background rate of Gator in comparison to other $\gamma$-ray screening HPGe detectors located in SURF [17], LNGS, Boulby [18], LSC [19], and LVdA [20]. Not all references provide the uncertainty in the count rate.  Detector | Location | Crystal | Efficiency | FWHM at | Rate
---|---|---|---|---|---
| (Overburden | Mass |  | $1332\text{\,}\mathrm{keV}$ | 60-2700 keV
| [m.w.e.]) | [kg] |  | [keV] | [counts/(kg$\cdot$day)]
Gator | LNGS (3600) | 2.2 | 100.5 % | 1.98 | $89.0\pm 0.7\text{\,}$
Maeve [21] | SURF (4300) | 2.0 |  185 %1. | 3.19 | 956.1
GeMPI 3 [22] | LNGS (3600) | 2.2 |  198.7 % | 2.20 | $24\pm 1.\text{\,}$
Belmont [21] | Boulby (2805) | 3.2 | 160 %1. | 1.92 | 135.0
GeOroel [19] | LSC (2450) | 2.2 | 109 %1. | 1.85 | 165.3
GeMSE [7, 23] | LVdA (620) | 2.0 | 107.7 % | 1.96 | $88\pm 1.\text{\,}$

##  4 Background analysis

After the detector upgrade and the cleaning of all shielding components, we
have acquired $74\text{\,}\mathrm{d}\mathrm{a}\mathrm{y}\mathrm{s}$ of
background data, shown in Figure 4. In the energy region between
$35\text{\,}\mathrm{k}\mathrm{e}\mathrm{V}2700\text{\,}\mathrm{k}\mathrm{e}\mathrm{V}$,
the background is dominated by $\gamma$-rays from detector and shielding
materials, primarily from the primordial 238U, 232Th, 235U and 40K. Of
particular concern is the gaseous 222Rn daughter from primordial 238U, which
is present in the laboratory environment. Although a dedicated ventilation
system in the underground laboratory helps to maintain a relatively low level
of radon in the experimental facilities [24], one of the motivations for the
upgrade, as described in section 2, was to tighten the enclosure so that the
$\mathrm{GN}_{2}$ purge displaces the environmental radon more effectively.

Figure 4: Recent Gator background spectrum (October – December 2021) using
$73.7\text{\,}\mathrm{d}\mathrm{a}\mathrm{y}\mathrm{s}$ of data. Prominent
$\gamma$-ray lines are labeled.

The integrated background rate is $82.0\pm
0.7\text{\,}\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{n}\mathrm{t}\mathrm{s}\mathrm{/}\mathrm{(}\mathrm{k}\mathrm{g}\cdot\mathrm{d}\mathrm{a}\mathrm{y}\mathrm{)}$
in the region
$100\text{\,}\mathrm{k}\mathrm{e}\mathrm{V}2700\text{\,}\mathrm{k}\mathrm{e}\mathrm{V}$,
as compared to the value of $102.8\pm
0.7\text{\,}\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{n}\mathrm{t}\mathrm{s}\mathrm{/}\mathrm{(}\mathrm{k}\mathrm{g}\cdot\mathrm{d}\mathrm{a}\mathrm{y}\mathrm{)}$
in 2010 [3]. We quote the rates for the region $60-2700$ keV in Table 1 to
allow for comparison with some of the most sensitive screening facilities.
Most of these present low count rates of $\mathcal{O}(100)$
$\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{n}\mathrm{t}\mathrm{s}\mathrm{/}\mathrm{(}\mathrm{k}\mathrm{g}\cdot\mathrm{d}\mathrm{a}\mathrm{y}\mathrm{)}$,
with the lowest count rate being for GeMPI 3 [22], followed by GeMSE [7] and
Gator. Also given in Table 1 are the detector locations and overburden in
meter water equivalent (m.w.e), the mass of each HPGe crystal, along with the
detector efficiencies and FWHM at
$1332\text{\,}\mathrm{k}\mathrm{e}\mathrm{V}$.

Table 2: Compton-subtracted rates in the 3$\sigma$-regions for selected $\gamma$-ray lines in the background spectrum. The rates of 3 different runs between 2010 and 2021 are shown, with live-times in days of 73.7 (2021), 98.5 (2015), and 51.4 (2010), respectively. | Rate [counts/day]
---|---
Energy [keV] | Chain/nuclide | October 2021 | October 2015 | April 2010 [3]
351.932 |  238U/214Pb | 0.41 $\pm$ 0.17 | 0.58 $\pm$ 0.14 | 0.7 $\pm$ 0.3
609.312 |  238U/214Bi | 0.26 $\pm$ 0.10 | 0.48 $\pm$ 0.09 | 0.6 $\pm$ 0.2
1120.29 |  238U/214Bi | < 0.28 | 0.18 $\pm$ 0.06 | 0.3 $\pm$ 0.1
1764.49 |  238U/214Bi | 0.14 $\pm$ 0.06 | 0.18 $\pm$ 0.05 | 0.08 $\pm$ 0.06
661.657 |  137Cs | 0.19 $\pm$ 0.09 | 0.24 $\pm$ 0.07 | 0.3 $\pm$ 0.1
1173.24 |  60Co | < 0.27 | < 0.34 | 0.5 $\pm$ 0.1
1332.51 |  60Co | < 0.21 | < 0.24 | 0.5 $\pm$ 0.1
1460.88 |  40K | 0.28 $\pm$ 0.08 | 0.43 $\pm$ 0.07 | 0.5 $\pm$ 0.1
2614.51 |  232Th/208Tl | 0.19 $\pm$ 0.05 | 0.22 $\pm$ 0.05 | 0.2 $\pm$ 0.1

Table 2 presents the integral count rates of $\gamma$-ray lines which are
typically present in the background data – these are lines from primordial
isotopes (or from their daughters) as well as from medium-lived cosmogenic or
anthropogenic isotopes which have a high branching ratio or detection
efficiency. The count rates of these $\gamma$-ray lines in the new background
data are consistent within the uncertainties or lower than the values measured
in 2010 [3]. As expected, the activities of cosmogenic radionuclides with
half-lives on the order of a few years, such as 60Co, have noticeably
decreased while underground. Although still consistent within errors, the
activities of 40K and isotopes from the 222Rn chain (e.g. 214Pb, 214Bi) seem
to have decreased as well, likely due to the cleaning of the detector cavity
and/or tightening of the enclosure.

##  5 Sample analysis and results

Before a sample is submitted to a radioassay in Gator, we evaluate whether its
mass, material, and expected radioactivity are likely to yield either values
within the sensitivity reach (after approximately
$30\text{\,}\mathrm{d}\mathrm{a}\mathrm{y}\mathrm{s}$ of screening) or upper
limits that fulfill the radiopurity requirements for the given sample. This
evaluation is performed by comparing the sample specifications to samples
previously measured in Gator and their achieved sensitivity. Once the sample
has been selected and measured in Gator, its data analysis is performed
following a counting method around the $\gamma$-ray lines of interest. This
section summarises this analysis procedure and presents an example of a
radioassay of photosensors conducted with the new background level.

### Analysis procedure

To estimate the activity of the samples, we perform a Geant4 simulation [25]
using a model of the exact geometry, chemical composition and density of
materials present in the sample, detector and cavity. The simulation of the
detector geometry and its components are validated using calibrated point and
extended sources, as described in detail in [3]. A source term for each
expected isotope or decay chain is distributed uniformly throughout the
sample. The decays and their subsequent interactions within the modeled
geometry are simulated via a Monte Carlo method in Geant4. The final energy
depositions within the detector are then stored and used to calculate the
efficiency of each gamma line. The number of simulated decays in the sample
volume is typically of $\mathcal{O}(10^{8})$ for each isotope of interest. The
ratio between the detected and the simulated number of gammas results in the
detection efficiency ($\varepsilon$). This efficiency is geometry-, material-,
and energy-dependent, as will be shown with the photosensor sample.

Background runs are acquired in between the materials screening, not longer
than 12 months before or after the sample measurement, with acquisition times
($t_{B}$) usually longer than that of the sample ($t_{S}$). Although the
samples are usually pre-purged in the load-lock chamber, the first hours/days
of data acquisition may present a higher rate due to an initially larger
concentration of 222Rn, which has a half-life of
$3.82\text{\,}\mathrm{d}\mathrm{a}\mathrm{y}\mathrm{s}$ and is still being
purged out of the cavity or emanating from the samples. The trigger rate can
be thus modeled with an exponential function during this initial period and
later reaches a constant level within uncertainties. This initial period, for
which the rate is dominated by an exponential behavior, as well as data in the
direct vicinity of $\mathrm{LN}_{2}$ refills (described in section 3), are
thus removed from the analysis data set. The acquired sample and background
data are then calibrated using the functions obtained with dedicated
calibration runs.

After the data selection and calibration, we count the number of events in a
$3\sigma$ region around the centroid of $\gamma$-ray lines of interest in both
the sample and the background data. The Compton background is estimated from
the left and right sidebands around each peak. These values are then used to
obtain the Compton-subtracted counts within each $\gamma$-ray peak in the
sample (S) and in the background (B) data. The latter is corrected for the
acquisition time in order to obtain the net signal $S_{net}$ from the sample
for a given $\gamma$-ray line:

$S_{net}=S-B\cdot\frac{t_{S}}{t_{B}}.$ (5.1)

The background and sample rates are usually low (with both S and B below 100
counts in the peak, as exemplified in Table 2); we thus use the method of
activities and upper limit determination for low-radioactivity measurements
detailed in [26]. We estimate the standard uncertainties on $x$ counts as
$\sqrt{x+1}$ and implement these in the calculation of the detection limit
$L_{d}$ which is defined as the level of true net signal that leads to a
certain probability of detection [26]. Following the recommendation of the
same reference, we report activity values if $S_{net}$ exceeds $L_{d}$, and
report upper limits on the activity otherwise. To translate $S_{net}$ from a
given $\gamma$-ray line to the specific activity of the isotope, we use the
following equation:

$A[\mathrm{Bq/kg}]=\frac{S_{net}}{r\cdot\varepsilon\cdot m\cdot t_{S}},$ (5.2)

where r is the branching ratio of the $\gamma$-ray line and m is the sample
mass. To calculate the error on the activity, we take both the error on
$S_{net}$ and on the efficiency into account, with the latter being a
$10\text{\,}\mathrm{\char 37\relax}$ systematic error (described in [3]). If
several lines of a given isotope are present in the spectrum, with
$S_{net}>L_{d}$, the activity is calculated as the error-weighted average.

### Photosensor sample

After the upgrade, we have screened two Hamamatsu R12699-406-M4
photomultiplier tubes (PMTs) [27], which are a viable candidate photosensor
for the future DARWIN detector. The baseline design of DARWIN features a
liquid-xenon target instrumented by two large arrays of photosensors [12]. The
sample photosensor has a square 2-inch photosensitive area, in contrast to the
circular 3-inch R11410 units used in XENON1T [28], XENONnT [29], LZ [30] and
PandaX-4T [31]. The square shape allows for a higher packing density in an
array as compared to circular tubes. Together with a higher photocathode
coverage of $75.0\text{\,}\mathrm{\char 37\relax}$, as opposed to
$61.8\text{\,}\mathrm{\char 37\relax}$ for R11410 PMTs at densest packing,
this implies an increase in the light collection efficiency of the detector.
Furthermore, the 2-inch sensor possesses a buoyancy that is two orders of
magnitude smaller in xenon. This feature suggests that an array of R12699 PMTs
would require a less rigid mechanical support structure and hence less
material close to the xenon target. As such, it would be feasible to reduce
the backgrounds that arise from detector materials if, in addition, the
average radioactivity of the R12699 unit can be reduced sufficiently.

Figure 5: (Left) Two Hamamatsu R12699-406-M4 PMTs inside the Gator sample
cavity. (Right) Geometry as implemented in the simulations. The upper part of
the copper cryostat around the HPGe crystal is shown in blue. For the PMTs,
the metal envelope and pins are colored red, the inner effective dynode
material orange, and the silica window together with the glass around the
pins, white. All other simulated components, such as the sample cavity, are
not shown for clarity.

Figure 5 shows the PMTs on the top of the detector as well as the geometry
implemented in the simulations of the detection efficiencies. The resulting
efficiencies for selected $\gamma$-ray lines are depicted in Figure 6. Below
approximately $300\text{\,}\mathrm{k}\mathrm{e}\mathrm{V}$, the detection
efficiency increases with energy as more gammas are able to penetrate the
sample itself as well as the copper cryostat and dead layer of the HPGe
crystal. At higher energies, it decreases again as energetic gammas will not
fully deposit their energy within the sensitive crystal volume. The finite
time resolution of the detector may lead to the registration of two temporally
unresolvable $\gamma$-ray depositions as a single summation peak. Gammas
resulting in a summation peak are hence missing in the peaks of their
individual energies which are consequently systematically reduced. For
example, the reduced efficiency at
$583.2\text{\,}\mathrm{k}\mathrm{e}\mathrm{V}$ is due to a summation peak at
$3197.7\text{\,}\mathrm{k}\mathrm{e}\mathrm{V}$ from the
$583.2\text{\,}\mathrm{k}\mathrm{e}\mathrm{V}$ and
$2614.5\text{\,}\mathrm{k}\mathrm{e}\mathrm{V}$ $\gamma$-ray lines of 208Tl.

Figure 6: Simulated detection efficiencies for selected lines in the sample
PMTs described in section 5. Values for a confinement of the impurities in the
simulations to the full PMT and only the glass around the pins, respectively,
are shown. The latter yields a systematically lower detection efficiency due
to geometrical and material absorption effects.

To demonstrate the impact of the assumptions in the efficiency simulations,
especially for complex geometries, we compare two different spatial
distributions of the decays: a homogeneous source confinement to the entire
PMT and an assumed localization of the radioimpurities to only the glass
around the pins (visible as white cylinders in Figure 5). A lower detection
efficiency is obtained for the latter, as shown in Figure 6, due to
geometrical and material absorption effects. This in turn would lead to higher
derived activities following Equation 5.2. Thus the spatial distribution of
radioimpurities is one of the factors included in the systematic uncertainty
of the final activity values.

Figure 7: Measured energy spectrum of the two Hamamatsu R12699 PMTs (orange)
compared to the background spectrum (blue). The background-subtracted spectrum
is given in red. Statistical uncertainties are indicated as transparent bands.
Prominent $\gamma$-ray lines from various isotopes are labeled.

The analysis of this sample is based on
$\sim$$41.7\text{\,}\mathrm{d}\mathrm{a}\mathrm{y}\mathrm{s}$ of data from the
screening of the two PMTs together with the
$\sim$$73.7\text{\,}\mathrm{d}\mathrm{a}\mathrm{y}\mathrm{s}$ of the
background reported in section 4. The spectra of these two data sets are shown
in Figure 7. Given in Table 3 are the resulting specific activities as
compared to Gator screening results for the Hamamatsu R11410-21 PMTs employed
in XENON1T and XENONnT [9] as well as the Hamamatsu R8520-06 PMTs used in
XENON100 [8]. These values are all based on detection efficiency simulations
assuming a source confinement to the entire PMT for consistent comparison. For
the isotopes 228Th, 60Co and 40K, the activity per active photocathode area of
the current R12699 model is lower compared to the R8520 PMTs [8]. For the
other isotopes listed in Table 3, the comparison is inconclusive, as only
upper limits are given for their values (particularly in the measurement of
the R8520 PMTs). With respect to the R11410-21 units [9], the sample PMT has
from equal to fourfold higher activities per active photocathode area. We
note, however, that the R12699 PMT is currently being optimized in terms of
low-activity materials. The eventually achievable activity, together with
further parameters, such as the performance of the PMT which is currently
being characterized, will decide on its applicability in DARWIN.

Table 3: Summary of activities for the main isotopes from the two Hamamatsu R12699 PMTs. Shown for comparison are the Gator results for the Hamamatsu R11410-21 PMTs used in XENON1T and XENONnT [9] and the Hamamatsu R8520-06 PMTs employed in XENON100 [8]. The activities are given per PMT (top) and per active photocathode area (bottom).  Isotope | R12699-406 | R11410-21 | R8520-06
---|---|---|---
Activity [mBq/PMT]
238U | < 6.1 |  8 $\pm$ 2 | < 15
226Ra |  0.60 $\pm$ 0.10 |  0.6 $\pm$ 0.1 | < 0.28
228Ra | < 0.65 |  0.7 $\pm$ 0.2 | < 0.59
228Th | < 0.53 |  0.6 $\pm$ 0.1 |  0.3 $\pm$ 0.1
235U | < 0.28 |  0.37 $\pm$ 0.09 | < 0.67
60Co |  1.31 $\pm$ 0.11 |  0.84 $\pm$ 0.09 |  0.60 $\pm$ 0.04
40K |  35 $\pm$ 4 |  12 $\pm$ 2 |  12.0 $\pm$ 0.8
137Cs | < 0.12 | – | < 0.10
Activity [mBq/cm2]
238U | < 0.260 |  0.25 $\pm$ 0.06 | < 3.569
226Ra |  0.026 $\pm$ 0.004 |  0.019 $\pm$ 0.003 | < 0.067
228Ra | < 0.028 |  0.022 $\pm$ 0.006 | < 0.140
228Th | < 0.023 |  0.019 $\pm$ 0.003 |  0.071 $\pm$ 0.017
235U | < 0.012 |  0.012 $\pm$ 0.003 | < 0.159
60Co |  0.055 $\pm$ 0.005 |  0.026 $\pm$ 0.003 |  0.144 $\pm$ 0.010
40K |  1.47 $\pm$ 0.16 |  0.37 $\pm$ 0.06 |  2.86 $\pm$ 0.18
137Cs | < 0.005 | – | < 0.024

This exemplary sample also demonstrates the sensitivity gain from the reduced
background rate. For instance, for the given background data, the counts
around the 214Bi line at $609.3\text{\,}\mathrm{k}\mathrm{e}\mathrm{V}$ exceed
the detection limit after a sample measurement time of about
$20\text{\,}\mathrm{d}\mathrm{a}\mathrm{y}\mathrm{s}$. At an overall
$25\text{\,}\mathrm{\char 37\relax}$ higher background (similar to the
integrated background rate in 2010 [3]), additional sample data taking of
$\sim$$15\text{\,}\mathrm{\char 37\relax}$ would be necessary. For lines with
longer required measurement times (due to a larger background at their energy,
lower detection efficiency or branching ratio, or lower activity), this effect
is even greater.

##  6 Summary and conclusions

In this work, we described an upgrade of the Gator HPGe $\gamma$-ray
spectrometry facility with respect to the detector enclosure and sample-
handling infrastructure. Following the improvements, a background rate of
$82.0\pm
0.7\text{\,}\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{n}\mathrm{t}\mathrm{s}\mathrm{/}\mathrm{(}\mathrm{k}\mathrm{g}\cdot\mathrm{d}\mathrm{a}\mathrm{y}\mathrm{)}$
in the
$100\text{\,}\mathrm{k}\mathrm{e}\mathrm{V}2700\text{\,}\mathrm{k}\mathrm{e}\mathrm{V}$
region was achieved over a 74-day counting period, reflecting a
$20\text{\,}\mathrm{\char 37\relax}$ decrease with respect to the previously
reported value [3]. This reduction can be attributed to the reduced activity
of the cosmogenic isotope 60Co over time
($\mathrm{T}_{1/2}=$5.3\text{\,}\mathrm{y}\mathrm{r}\mathrm{s}$$) and also
likely due to the cleaning of the detector cavity and tightening of the
enclosure. Further, the new enclosure improved the handling of samples,
reducing the risk of introducing contamination into the cavity.

The HPGe detector has been under stable operation for over a decade, ensured
by constant monitoring of its operating parameters as well as the facility
environment. It undergoes regular calibrations, which allow us to monitor the
energy reconstruction, resolution, and efficiency. Recent calibrations yield
an energy resolution of $2.03\pm 0.04\text{\,}\mathrm{k}\mathrm{e}\mathrm{V}$
at $1332\text{\,}\mathrm{k}\mathrm{e}\mathrm{V}$ (FWHM).

We summarised the analysis procedure, optimised for ultra-low radioactivity
samples. This analysis relies on the detection efficiency for each specific
radioisotope and sample, estimated by means of Geant4 simulations of the
precise detector and sample geometries. As an example from the ongoing DARWIN
radioassay campaign, we report results from a 42-day screening of the
Hamamatsu R12699-406-M4 PMT, a possible alternative to the photosensors
currently used by some of the most sensitive dark matter direct detectors.

Gator was previously used for the XENON dark matter and GERDA, LEGEND-200
neutrinoless double beta decay experiments and will be used in the future for
LEGEND-1000 and for DARWIN. These experiments target yet lower backgrounds and
hence more radiopure materials, which in turn requires even higher
sensitivities in their radioassay, that could be achieved in Gator thanks to
the reduced background rate.

## Acknowledgments

This work was supported by the Swiss National Science Foundation under Grant
No. 200020-188716, by the Candoc Grant No. K-72312-09-01 from the University
of Zurich, by the European Unions Horizon 2020 research and innovation
programme under the Marie Sklodowska-Curie grant agreements No. 690575 and No.
674896, and by the European Research Council (ERC) under the European Union’s
Horizon 2020 research and innovation programme, grant agreement No. 742789
(Xenoscope). We thank Andreas James, Francesco Piastra, Roberto Corrieri, and
Junji Naganoma for their valuable contributions to the facility upgrade as
well as Diego Ramírez Garcia for his careful review of this manuscript. We
also thank the mechanical workshop in the Physics Department for their
continuous support and the LNGS mechanical and electrical technicians for the
support provided onsite.

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# Minimal hypersurfaces in $\mathbb{S}^{5}$ with constant scalar curvature and
zero Gauss curvature are totally geodesic

Qing Cui  School of Mathematics
Southwest Jiaotong University
611756 Chengdu
Sichuan
China<EMAIL_ADDRESS>
###### Abstract.

We show that a closed minimal hypersurface in $\mathbb{S}^{5}$ with constant
scalar curvature and zero Gauss curvature is totally geodesic.

##  1\. Introduction

Let $M$ be a minimal hypersurface in $\mathbb{S}^{n+1}$ with second
fundamental form $A$. In 1968, Simons [29] derived the famous Simons’
identity:

(1) $\displaystyle\dfrac{1}{2}\Delta\left\lvert
A\right\rvert^{2}=\left\lvert\nabla A\right\rvert^{2}+\left(n-\left\lvert
A\right\rvert^{2}\right)\left\lvert A\right\rvert^{2},$

where $\nabla A$ is the covariant derivation of $A$. Therefore, if $M$ is
closed and assume $0\leq\left\lvert A\right\rvert^{2}\leq n$, then the
divergence theorem gives $\left\lvert A\right\rvert^{2}\equiv 0$ or
$\left\lvert A\right\rvert^{2}\equiv n$. This pinching phenomenon has
attracted many mathematicians’ attention. Chern-do Carmo-Kobayashi [9] and
Lawson [19] independently characterized the case of $\left\lvert
A\right\rvert^{2}\equiv n$: the Clifford hypersurfaces are the only minimal
hypersurfces in $\mathbb{S}^{n+1}$ with $\left\lvert A\right\rvert^{2}\equiv
n$. Assume $R_{M}$ is the scalar curvature of $M$. By the Gauss equation, one
has $\left\lvert A\right\rvert^{2}=n(n-1)+R_{M}$. Therefore, $M$ has constant
scalar curvature iff $\left\lvert A\right\rvert^{2}$ is constant. The authors
[9] proposed the following conjecture:

######  Conjecture 1.

Let $M^{n}$ be a closed immersed minimal hypersurface of the unit sphere
$\mathbb{S}^{n+1}$ with constant scalar curvature $R_{M}$. Then for each $n$,
the set of all possible values for $R_{M}$ is discrete.

Now this conjecture is known as Chern conjecture. In 1983, Peng and Terng [26,
27] made the first breakthrough towards Chern Conjecture, they showed: _If
$\left\lvert A\right\rvert^{2}>n$, then $\left\lvert
A\right\rvert^{2}>n+\frac{1}{12n}$. Moreover, for $n=3$, $\left\lvert
A\right\rvert^{2}\geq 6$ if $\left\lvert A\right\rvert^{2}>3$. _ In 1993,
Chang [7] completed the proof of Chern Conjecture for $n=3$. After that, Yang-
Cheng [36] and Suh-Yang [30] improved the constant from $\frac{1}{12n}$ to
$\frac{3n}{7}$. Based on known examples, mathematicians believe this constant
should be $n$, that is, if $\left\lvert A\right\rvert^{2}\equiv const.>n$,
then $\left\lvert A\right\rvert^{2}\geq 2n$ (this is usually called the second
pinching conjecture). Up to now, even the second pinching conjecture is still
open for higher dimension.

Note that, in Simons’ pinching result, $\left\lvert A\right\rvert^{2}$ is not
required to be constant. Thus, many mathematicians believe the second pinching
conjecture still holds without the assumption that $\left\lvert
A\right\rvert^{2}$ is constant. In this case, Peng-Terng [26, 27] showed that
there does exist a pinching phenomenon if $\left\lvert A\right\rvert^{2}>n$
and $n\leq 5$. Later, Cheng-Ishikawa [8], Wei-Xu [34] and Zhang [37] promoted
it to $n\leq 8$. Finally, Ding-Xin [16] proved that if $\left\lvert
A\right\rvert^{2}>n$ then $\left\lvert A\right\rvert^{2}>n+\frac{n}{23}$ for
all dimension $n$. After that, Xu-Xu [35] and Lei-Xu-Xu [20] improved the
pinching constant $\frac{n}{23}$ to $\frac{n}{18}$.

Since all the known minimal hypersurfaces in $\mathbb{S}^{n+1}$ with constant
scalar curvature are isoparametric (equivalently to have constant principal
curvatures), mathematicians proposed the following strong version of Chern
conjecture:

######  Conjecture 2.

A closed immersed minimal hypersurface $M^{n}$ of $\mathbb{S}^{n+1}$ with
constant scalar curvature must be isoparametric.

Actually, if $M$ is a minimal isoparametric hypersurface in
$\mathbb{S}^{n+1}$, according to Münzner’s result [25], $\left\lvert
A\right\rvert^{2}$ can only take values in the set
$\left\\{0,n,2n,3n,5n\right\\}$. It is worth pointing out that, isoparametric
hypersurfaces in $\mathbb{S}^{n+1}$ are completely classified until recently
(see [6, 10, 11, 12, 17, 18, 23, 24]). When $n=3$, Chang [7] gave an
affirmative answer to Conjecture 2. When $n=4$, Deng-Gu-Wei [15] proved
Conjecture 2 with an additional assumption that $M^{4}$ is Willmore. Note that
in this case, $M$ is Willmore iff $f_{3}\equiv 0$ (see [22]), where $f_{r}$
denotes the $r$-th power sum of the principal curvatures for each positive
integer $r$. With assumption that $R_{M}\geq 0$, $f_{3}$ and the number of
distinct principal curvatures $g$ are constant, Tang and Yang [31] proved
Conjecture 2. Recently, Li [21] showed Conjecture 2 is true if $f_{3}$ is
constant satisfies $f_{3}^{2}\leq 72$ and the Gauss curvature $\mathcal{K}$
satisfies $\mathcal{K}\leq 1$. For higher dimensional case, with assumptions
that the Gauss curvature $\mathcal{K}$ is constant and $M$ has three pairwise
distinct principal curvatures everywhere, de Almeida-Brito-Scherfner-Weiss
[14] showed that Conjecture 2 is true. Recently, Tang-Wei-Yan [32] and Tang-
Yan [33] showed Conjecture 2 is true if $f_{r}$ is constant for each
$k=1,\cdots,n-1$ and $R_{M}\geq 0$. Their results strongly support Conjecture
2 and generalized de Almeida-Brito’s results in [13].

In this paper, we focus on the case of dimension $n=4$ and $M$ has vanishing
Gauss curvature. In this case, a minimal isoparametric hypersurface must be
totally geodesic (see Lemma 3 in Section 3). Therefore, based on the
Conjecture 2, a natural question is: Is a closed minimal hypersurface with
constant scalar curvature and zero Gauss curvature totally geodesic? We give
an affirmative answer to this question as follows:

######  Theorem 1.

Let $M$ be a closed minimal hypersurface in $\mathbb{S}^{5}$ with constant
scalar curvature and zero Gauss curvature, then $M$ is totally geodesic.

We emphasize that without the assumption of constant scalar curvature, there
are many non-totally geodesic minimal hypersurfaces in a unit sphere with zero
Gauss curvature, cf. [28].

This paper is arranged as follows. In Section 2, we set notations and review
some known formulas and results. In Section 3, we give the proof of Theorem 1.

##  2\. Preliminary

In this section, we will fix some notations and list several known formulas.

Let $M$ be a minimal hypersurface in $\mathbb{S}^{5}$ with second fundamental
form $A$. We choose a local orthonormal frame
$\left\\{e_{i}\right\\}_{i=1}^{5}$ in $\mathbb{S}^{5}$ such that
$e_{1},\cdots,e_{4}$ are tangent to $M$. Let
$\left\\{\omega^{i}\right\\}_{i=1}^{5}$ be the dual frame of
$\left\\{e_{i}\right\\}_{i=1}^{5}$. Then $A$ can be written as
$A=\sum_{i,j=1}^{4}h_{ij}\omega^{i}\otimes\omega^{j}$ with $h_{ij}=h_{ji}$.
Unless otherwise specified, throughout this paper, the summation is always
from 1 to 4. The minimality of $M$ implies $0=H=\dfrac{1}{4}\sum_{i}h_{ii}$.
Set

$\displaystyle F_{,i}=\nabla_{i}F,\ \ F_{,ij}=\nabla_{j}\nabla_{i}F,\ \
h_{ijk}=\nabla_{k}h_{ij}\ \text{and}\ h_{ijkl}=\nabla_{l}\nabla_{k}h_{ij},$

where $\nabla_{j}$ is the covariant differentiation operator with respect to
$e_{j}$. The Gauss equation, the Codazzi equation and the Ricci formulas are
given by (cf. [9])

(2) $\displaystyle R_{ijkl}=$
$\displaystyle\delta_{ik}\delta_{jl}-\delta_{il}\delta_{jk}+h_{ik}h_{jl}-h_{il}h_{jk}.$
(3) $\displaystyle h_{ijk}=$ $\displaystyle h_{ikj}.$ (4) $\displaystyle
h_{ijkl}-h_{ijlk}=$
$\displaystyle\sum_{m}h_{im}R_{mjkl}+\sum_{m}h_{mj}R_{mikl}.$

From the Gauss equation (2), it follows that
$R_{M}=\sum_{i,j}R_{ijij}=12-\left\lvert A\right\rvert^{2}.$ Therefore,
$\left\lvert A\right\rvert^{2}$ is constant iff the scalar curvature is
constant. If $M$ has constant scalar curvature, Simons’ identity (1) implies

(5) $\displaystyle\left\lvert\nabla A\right\rvert^{2}=\left\lvert
A\right\rvert^{2}\left(\left\lvert A\right\rvert^{2}-4\right).$

Assume $A$ has principal curvature $\lambda_{1},\cdots,\lambda_{4}$ at some
point. Combining (2) with (4) gives

(6) $\displaystyle
h_{ijkl}-h_{ijlk}=\left(\lambda_{i}-\lambda_{j}\right)\left(1+\lambda_{i}\lambda_{j}\right)\left(\delta_{ik}\delta_{jl}-\delta_{il}\delta_{jk}\right).$

For each positive integer $k$, denote by

(7) $\displaystyle f_{k}={\rm
tr}A^{k},\quad\sigma_{k}=\sum_{i_{1}<\cdots<i_{k}}\lambda_{i_{1}}\cdots\lambda_{i_{k}}.$

Although $f_{k}$ and $\sigma_{k}$ are defined by using the principal
curvatures, they are actually globally well defined smooth functions. A direct
computation gives,

(8) $\displaystyle\begin{cases}&f_{1}=\sigma_{1}=4H=0,\\\
&f_{2}=\sigma_{1}^{2}-2\sigma_{2}=\left\lvert A\right\rvert^{2},\\\
&f_{3}=\sigma_{1}^{3}-3\sigma_{1}\sigma_{2}+3\sigma_{3}=3\sigma_{3},\\\
&f_{4}=\sigma_{1}^{4}-4\sigma_{1}^{2}\sigma_{2}+4\sigma_{1}\sigma_{3}+2\sigma_{2}^{2}-4\sigma_{4}=\frac{1}{2}\left\lvert
A\right\rvert^{4}-4\mathcal{K},\end{cases}$

where $\mathcal{K}=\sigma_{4}=\lambda_{1}\lambda_{2}\lambda_{3}\lambda_{4}$ is
the Gauss curvature (also called Gauss-Kronecker curvature). Note that

(9) $\displaystyle\mathcal{K}\equiv 0\ \ \text{iff}\ \
f_{4}\equiv\dfrac{1}{2}\left\lvert A\right\rvert^{4}.$

The following identities are due to Peng and Terng [26],

(10) $\displaystyle\Delta f_{3}=$ $\displaystyle 3\left(4-\left\lvert
A\right\rvert^{2}\right)f_{3}+6\mathscr{C},$ (11) $\displaystyle\Delta f_{4}=$
$\displaystyle 4\left(4-\left\lvert
A\right\rvert^{2}\right)f_{4}+4\left(2\mathscr{A}+\mathscr{B}\right),$

where

(12)
$\displaystyle\mathscr{A}=\sum_{i,j,k}\lambda_{i}^{2}h_{ijk}^{2},\quad\mathscr{B}=\sum_{i,j,k}\lambda_{i}\lambda_{j}h_{ijk}^{2},\quad\mathscr{C}=\sum_{i,j,k}\lambda_{i}h_{ijk}^{2}.$

##  3\. Proof of the main theorem

The following result is due to Cartan [5].

######  Lemma 2 (cf. [3, page 84]).

Let $M$ be an isoparametric hypersurface in $\mathbb{S}^{n+1}$. Denote by $g$
the number of distinct principal curvatures, and by $\mu_{1},\cdots,\mu_{g}$
the distinct principal curvatures with corresponding multiplicities
$m_{1},\cdots,m_{g}$. Then for each fixed $i=1,\cdots,g$,

(13) $\displaystyle\sum_{j=1,\ j\neq
i}^{g}m_{j}\dfrac{1+\mu_{i}\mu_{j}}{\mu_{i}-\mu_{j}}=0.$

Although isopermetric hypersurfaces are completely classified, we can give a
direct proof of the following result by Cartan’s fundamental formula.

######  Lemma 3.

Assume $M$ is a minimal isoparametric hypersurface in $\mathbb{S}^{5}$ with
zero Gauss curvature, then $M$ is totally geodesic.

###### Proof.

Assume $\lambda_{1}\leq\lambda_{2}\leq\lambda_{3}\leq\lambda_{4}$ are the
principal curvatures of $M$. By our assumption,

$\displaystyle 0=4H=\sum_{i=1}^{4}\lambda_{i},\quad
0=\mathcal{K}=\lambda_{1}\lambda_{2}\lambda_{3}\lambda_{4}.$

Suppose $M$ is not totally geodesic, we can assume, without loss of
generality, $\lambda_{2}=0$. Then the principal curvatures are

$\displaystyle\lambda_{1}<\lambda_{2}=0\leq\lambda_{3}\leq\lambda_{4}>0.$

There are three possibilities:

  * •

$\lambda_{3}=0$. In this case, we have

$\displaystyle
g=3,\mu_{1}=\lambda_{1},\mu_{2}=0,\mu_{3}=\lambda_{4}=-\lambda_{1},m_{1}=1,m_{2}=2,m_{3}=1.$

Therefore, for $i=1$ in (13), we have

$\displaystyle\dfrac{2}{\lambda_{1}}+\dfrac{1-\lambda_{1}^{2}}{2\lambda_{1}}=0,$

which implies $\lambda_{1}=-\sqrt{5}$. Consequently, $\left\lvert
A\right\rvert^{2}=\lambda_{1}^{2}+\lambda_{4}^{2}=10$. However, from Münzner’s
result [25], $\left\lvert A\right\rvert^{2}=(g-1)n=2\times 4=8$, we get a
contradiction.

  * •

$\lambda_{3}=\lambda_{4}>0$. In this case, we have

$\displaystyle
g=3,\mu_{1}=\lambda_{1},\mu_{2}=0,\mu_{3}=\lambda_{3}=\lambda_{4}=-\dfrac{1}{2}\lambda_{1},m_{1}=1,m_{2}=1,m_{3}=2.$

Therefore, for $i=2$ in (13), we obtain,

$\displaystyle
0=\dfrac{1}{0-\lambda_{1}}+2\dfrac{1}{0-\left(-\frac{1}{2}\lambda_{1}\right)}=\dfrac{3}{\lambda_{1}},$

which is impossible.

  * •

$0<\lambda_{3}<\lambda_{4}$. In this case, we have

$\displaystyle g=4,\mu_{i}=\lambda_{i},m_{i}=1,i=1,\cdots,4.$

Therefore, for $i=2$ in (13), we obtain,

$\displaystyle
0=\dfrac{1}{0-\lambda_{1}}+\dfrac{1}{0-\lambda_{3}}+\dfrac{1}{0-\lambda_{4}}=-\dfrac{\lambda_{1}\lambda_{3}+\lambda_{1}\lambda_{4}+\lambda_{3}\lambda_{4}}{\lambda_{1}\lambda_{3}\lambda_{4}},$

which implies
$\lambda_{1}\lambda_{3}+\lambda_{1}\lambda_{4}+\lambda_{3}\lambda_{4}=0$.
Since $\lambda_{1}+\lambda_{3}+\lambda_{4}=0,$ we have

$\displaystyle\lambda_{3}^{2}+\lambda_{3}\lambda_{4}+\lambda_{4}^{2}=0,$

which is impossible for $\lambda_{4}>\lambda_{3}>0$.

Therefore, $M$ must be totally geodesic. ∎

We also need an elementary algebraic lemma.

######  Lemma 4.

Assume $a,b,c\in\mathbb{R}$ satisfying $a\leq b\leq c$ and $a+b+c=0$, then

$\displaystyle-\dfrac{1}{\sqrt{6}}\left(a^{2}+b^{2}+c^{2}\right)^{3/2}\leq
a^{3}+b^{3}+c^{3}\leq\dfrac{1}{\sqrt{6}}\left(a^{2}+b^{2}+c^{2}\right)^{3/2},$

and

$\displaystyle a^{3}+b^{3}+c^{3}=$
$\displaystyle-\dfrac{1}{\sqrt{6}}\left(a^{2}+b^{2}+c^{2}\right)^{3/2}\ \
\text{iff}\ \ b=c=-\dfrac{a}{2},$ $\displaystyle a^{3}+b^{3}+c^{3}=$
$\displaystyle\dfrac{1}{\sqrt{6}}\left(a^{2}+b^{2}+c^{2}\right)^{3/2}\ \
\text{iff}\ \ a=b=-\dfrac{c}{2},$ $\displaystyle a^{3}+b^{3}+c^{3}=$
$\displaystyle 0\ \ \text{iff}\ \ b=0\ \ \text{and}\ \ a=-c.$

######  Lemma 5.

Let $M$ be a closed minimal hypersurface in $\mathbb{S}^{5}$ with constant
scalar curvature and zero Gauss-Kronecker curvature. If the smallest (or
largest) principal curvature $\lambda_{1}$ has multiplicity two at some point
$p$, then $M$ is totally geodesic.

###### Proof.

Assume $M$ is not totally geodesic, but the multiplicity of the smallest
principal curvature $\lambda_{1}$ is two at $p$. Since $\mathcal{K}=0$, then
at $p$ we can assume the principal curvatures are

(14)
$\displaystyle\lambda_{1}=\lambda_{2}=-\lambda<0=\lambda_{3}<\lambda_{4}=2\lambda>0.$

At the point $p$, by Lemma 4, $f_{3}=\frac{1}{\sqrt{6}}\left\lvert
A\right\rvert^{3}$ attains $f_{3}$’s maximum. Therefore at $p$, we have for
each $k$

(15) $\displaystyle\begin{cases}&H_{,k}=0;\\\ &\left(\left\lvert
A\right\rvert^{2}\right)_{,k}=0;\\\ &\left(f_{3}\right)_{,k}=0;\\\
&\left(f_{4}\right)_{,k}=0,\end{cases}\ \ \text{which yields}\ \
\begin{cases}&\sum_{i}h_{iik}=0,\\\ &\sum_{i}\lambda_{i}h_{iik}=0,\\\
&\sum_{i}\lambda_{i}^{2}h_{iik}=0,\\\
&\sum_{i}\lambda_{i}^{3}h_{iik}=0,\end{cases}$

Consider that the value of principal curvatures (14), we have

$\displaystyle h_{11k}+h_{22k}=0,\ \ h_{44k}=h_{33k}=0.$

We have

$\displaystyle\left\lvert\nabla A\right\rvert^{2}=$ $\displaystyle
4\left(h_{111}^{2}+h_{112}^{2}\right)+6\left(h_{113}^{2}+h_{114}^{2}+h_{123}^{2}+h_{124}^{2}+h_{134}^{2}+h_{234}^{2}\right).$
$\displaystyle\mathscr{A}=$
$\displaystyle\sum_{i,j,k}\lambda_{i}^{2}h_{ijk}^{2}$ $\displaystyle=$
$\displaystyle\lambda^{2}\sum_{j,k}\left(h_{1jk}^{2}+h_{2jk}^{2}+4h_{4jk}^{2}\right)$
$\displaystyle=$ $\displaystyle\dfrac{1}{6}\left\lvert
A\right\rvert^{2}\left[4\left(h_{111}^{2}+h_{112}^{2}+h_{113}^{2}+h_{123}^{2}\right)+10\left(h_{234}^{2}+h_{134}^{2}\right)+12\left(h_{114}^{2}+h_{124}^{2}\right)\right].$
$\displaystyle\mathscr{B}=$
$\displaystyle\sum_{i,j,k}\lambda_{i}\lambda_{j}h_{ijk}^{2}$ $\displaystyle=$
$\displaystyle\lambda^{2}\sum_{k}\left(h_{11k}^{2}+h_{22k}^{2}+2h_{12k}^{2}-4h_{14k}^{2}-4h_{24k}^{2}\right)$
$\displaystyle=$
$\displaystyle\lambda^{2}\left[2\left(h_{113}^{2}+h_{123}^{2}\right)+4\left(h_{111}^{2}+h_{112}^{2}-h_{134}^{2}-h_{234}^{2}\right)-6\left(h_{114}^{2}+h_{124}^{2}\right)\right].$

Therefore, combined the above formula with (11), we have,

$\displaystyle
3\left[4\left(h_{111}^{2}+h_{112}^{2}\right)+6\left(h_{113}^{2}+h_{114}^{2}+h_{123}^{2}+h_{124}^{2}+h_{134}^{2}+h_{234}^{2}\right)\right]$
$\displaystyle=$ $\displaystyle 3\left\lvert\nabla A\right\rvert^{2}$
$\displaystyle=$ $\displaystyle 3\left\lvert
A\right\rvert^{2}\left(\left\lvert A\right\rvert^{2}-4\right)$
$\displaystyle=$
$\displaystyle\dfrac{1}{\lambda^{2}}\left(2\mathscr{A}+\mathscr{B}\right)$
$\displaystyle=$ $\displaystyle
12\left(h_{111}^{2}+h_{112}^{2}\right)+10\left(h_{113}^{2}+h_{123}^{2}\right)+16\left(h_{234}^{2}+h_{134}^{2}\right)+18\left(h_{114}^{2}+h_{124}^{2}\right),$

which implies

$\displaystyle
0=8\left(h_{113}^{2}+h_{123}^{2}\right)+2\left(h_{134}^{2}+h_{234}^{2}\right).$

Consequently,

$\displaystyle h_{113}=h_{123}=h_{134}=h_{234}=0.$

Moreover, since $H,\left\lvert A\right\rvert^{2}$ and $f_{4}$ are all
constants, we have, for all $1\leq k,l\leq 4$,

$\displaystyle\begin{cases}&H_{,kl}=0,\\\ &\left(\left\lvert
A\right\rvert^{2}\right)_{,kl}=0,\\\ &\left(f_{4}\right)_{,kl}=0.\end{cases}$

A direct computation yields,

(16) $\displaystyle\begin{cases}&\sum_{i}h_{iikl}=0,\\\
&\sum_{i,j}\left(h_{ijk}h_{ijl}+h_{ij}h_{ijkl}\right)=0,\\\
&\sum_{i}\left(\lambda_{i}^{3}h_{iikl}\right)+\sum_{i,j}\left(2\lambda_{i}^{2}h_{ijk}h_{ijl}+\lambda_{i}\lambda_{j}h_{ijk}h_{ijl}\right)=0.\end{cases}$

Note that, at the point $p$, we have (14) and

$\displaystyle
h_{11k}+h_{22k}=0,h_{33k}=h_{44k}=0,h_{113}=h_{123}=h_{134}=h_{234}=0.$

Therefore, for $k=l=3$, (16) becomes,

$\displaystyle\begin{cases}&h_{1133}+h_{2233}+h_{3333}+h_{4433}=0,\\\
&-\lambda h_{1133}-\lambda h_{2233}+2\lambda h_{4433}=0,\\\
&-\lambda^{3}h_{1133}-\lambda^{3}h_{2233}+8\lambda^{3}h_{4433}=0,\end{cases}$

which implies $h_{4433}=0$. While for $k=l=4$, (16) becomes,

$\displaystyle\begin{cases}&h_{1144}+h_{2244}+h_{3344}+h_{4444}=0,\\\
&-\lambda h_{1144}-\lambda h_{2244}+2\lambda
h_{4444}=-2\left(h_{114}^{2}+h_{124}^{2}\right),\\\
&-\lambda^{3}h_{1144}-\lambda^{3}h_{2244}+8\lambda^{3}h_{4444}=-6\lambda^{2}\left(h_{114}^{2}+h_{124}^{2}\right),\end{cases}$

which implies $h_{3344}=0$. Consequently,

$\displaystyle h_{3344}-h_{4433}=0.$

However, (6) and (14) yield

$\displaystyle h_{3344}-h_{4433}=-2\lambda<0.$

We get a contradiction. ∎

In dimension four, the Gauss-Bonnet-Chern formula is quite special. In our
case, we have the following lemma.

######  Lemma 6.

Assume $M$ is a closed minimal hypersurface in $\mathbb{S}^{5}$ with second
fundamental form $A$, Euler number $\chi(M)$, and $f_{4}$ is defined by (7).
Then the Gauss-Bonnet-Chern formula becomes

(17) $\displaystyle\int_{M}\left(\dfrac{3}{2}\left\lvert
A\right\rvert^{4}-3f_{4}-2\left\lvert
A\right\rvert^{2}+12\right)=16\pi^{2}\chi\left(M\right).$

###### Proof.

For a closed 4-dimensional manifold, the Gauss-Bonnet-Chern formula (see [2]
or [4]) reads

(18) $\displaystyle\int_{M}\left(\frac{R_{M}^{2}}{3}-\left\lvert\rm
Ric\right\rvert^{2}+\frac{\left\lvert
W\right\rvert^{2}}{2}\right)=16\pi^{2}\chi(M),$

where $R_{M},Ric,W$ are the scalar curvature, Ricci tensor, Weyl curvature
tensor of $M$ respectively, which can be calculated as follows (see [1,
p.117]),

$\displaystyle R_{M}=$ $\displaystyle\sum_{i,j}R_{ijij},$
$\displaystyle\left\lvert{\rm Ric}\right\rvert^{2}=$
$\displaystyle\sum_{i,j}R_{ij}^{2}=:\sum_{i,j}\left(\sum_{k}R_{ikjk}\right)^{2},$
$\displaystyle\left\lvert W\right\rvert^{2}=$
$\displaystyle\sum_{i,j,k,l}W_{ijkl}^{2}$ $\displaystyle=$
$\displaystyle\sum_{i,j,k,l}\left[R_{ijkl}-\dfrac{1}{2}\left(R_{ik}\delta_{jl}-R_{il}\delta_{jk}+R_{jl}\delta_{ik}-R_{jk}\delta_{il}\right)+\dfrac{R_{M}}{6}\left(\delta_{ik}\delta_{jl}-\delta_{il}\delta_{jk}\right)\right]^{2}.$

Then combined the above formulas and Gauss equation (2), a direct computation
yields

$\displaystyle R_{M}=$ $\displaystyle 12-\left\lvert A\right\rvert^{2},$
$\displaystyle\left\lvert Ric\right\rvert^{2}=$ $\displaystyle 36-6\left\lvert
A\right\rvert^{2}+f_{4},$ $\displaystyle\left\lvert W\right\rvert^{2}=$
$\displaystyle\dfrac{7}{3}\left\lvert A\right\rvert^{4}-4f_{4}.$

Substituting the above formulas into Gauss-Bonnet-Chern formula (17), we
obtain (18). ∎

Remark. Since for $n\geq 4$, locally conformally flat is equivalent to
$W\equiv 0$. Therefore, by the proof of the above lemma, a minimal
hypersurface in $\mathbb{S}^{5}$ is locally conformally flat iff $\left\lvert
A\right\rvert^{4}=\dfrac{12}{7}f_{4}$. By a direct computation, the trace-free
Ricci tensor $\mathring{Ric}=Ric-\dfrac{R_{M}}{4}Id$ satisfies

$\displaystyle\left\lvert\mathring{Ric}\right\rvert^{2}=f_{4}-\dfrac{1}{4}\left\lvert
A\right\rvert^{4}.$

Hence, $M$ is Einstein iff $\mathring{Ric}\equiv 0$, i.e., $4f_{4}=\left\lvert
A\right\rvert^{4}$.

We also need a result due to Cheng-Yang.

######  Lemma 7 ([36, Theorem 2]).

Let $M^{4}$ be a closed minimal hypersurface of $S^{5}$ with constant scalar
curvature and constant $f_{4}$. If $\left\lvert A\right\rvert^{2}>4$, then
$\left\lvert A\right\rvert^{2}\geq\frac{20}{3}$.

Remark. Cheng-Yang actually proved the above lemma for general dimension $n$
and with the assumption that $f_{3}$ is constant instead of $f_{4}$ is
constant. The key ingredients of their proof are equalities (10) and (11).
When $f_{4}$ is constant, we can change a little in the proof of [36, Lemma
3.1], such that

$\displaystyle 0=$ $\displaystyle 3\left(4-\left\lvert
A\right\rvert^{2}\right)f_{3}+6\mathscr{C},\ \ \text{at a point}\ \ x_{0}\in
M,$ $\displaystyle 0=$ $\displaystyle 4\left(4-\left\lvert
A\right\rvert^{2}\right)f_{4}+4\left(2\mathscr{A}+\mathscr{B}\right),\ \
\text{at all points in }\ M.$

Consider at $x_{0}$, we can get the same result as [36, Lemma 3.1], and
consequently we can obtain Lemma 7. Cheng-Yang also mentioned this lemma in
[36, Remark 1].

Now we can give the proof of Theorem 1.

###### Proof of Theorem 1.

Suppose $M$ is not totally geodesic, then from Lemma 5, we know the smallest
principal curvature $\lambda_{1}$ can not has multiplicity two at any point.
Since $\mathcal{K}=\lambda_{1}\lambda_{2}\lambda_{3}\lambda_{4}=0$,
$\lambda_{1}$ can not has multiplicity 3 at any point either. Therefore,
$\lambda_{1}$ always has multiplicity one. Consequently, the unit eigenvectors
of $\lambda_{1}$ form a nowhere vanishing vector field on $M$, which implies
$\chi(M)=0$ by the Poincaré-Hopf theorem. Combined this fact with
$\mathcal{K}\equiv 0$ and relation (9), the Gauss-Bonnet-Chern formula (17) is
reduced to

$\displaystyle\int_{M}\left(\left\lvert A\right\rvert^{2}-6\right)=0,$

which implies $\left\lvert A\right\rvert^{2}=6$.

On the other hand, since $M$ has constant scalar curvature Simons’ identity
implies $\left\lvert A\right\rvert^{2}\geq 4$. However, $\left\lvert
A\right\rvert^{2}=4$ iff $M$ is a Clifford hypersurface ([9, 19]), which is
impossible since the Clifford hypersurfaces have non-zero Gauss curvature.
Thus, we have $\left\lvert A\right\rvert^{2}>4$. Notice that, $\mathcal{K}=0$
implies $f_{4}=\frac{1}{2}\left\lvert A\right\rvert^{4}$ is constant.
Consequently, we have $\left\lvert A\right\rvert^{2}\geq\frac{20}{3}$ from
Lemma 7. We get a contradiction. ∎

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