[1]The RPC ECOgas@GIF++ Collaboration: L. Quaglia [2,3]D. Ramos

1]INFN Sezione di Torino, Via P. Giuria 1, 10126 Torino, Italy 2]Politecnico
di Bari, Dipartimento Interateneo di Fisica, via Amendola 173, 70125 Bari,
Italy 3]INFN Sezione di Bari, Via E. Orabona 4, 70125 Bari, Italy 4]INFN -
Laboratori Nazionali di Frascati, Via Enrico Fermi 54, 00044 Frascati (Roma),
Italy 5]Sapienza Università di Roma, Dipartimento di Ingegneria Chimica
Materiali Ambiente, Piazzale Aldo Moro 5, 00185 Roma, Italy 6]Università degli
studi di Bari, Dipartimento Interateneo di Fisica, via Amendola 173, 70125
Bari, Italy 7]INFN Sezione di Bologna, Viale C. Berti Pichat 4/2, 40127
Bologna, Italy 8]Helwan University, Helwan Sharkeya, Helwan, Cairo Governorate
4037120, Egypt 9]INFN Sezione di Pavia, Via A. Bassi 6, 27100 Pavia, Italy
10]Ghent University, Dept. of Physics and Astronomy, Proeftuinstraat 86,
B-9000 Ghent, Belgium 11]Universidade do Estado do Rio de Janeiro, R. São
Francisco Xavier, 524 - Maracanã, Rio de Janeiro - RJ, 20550-013, Brasil
12]Università degli studi di Torino, Dipartimento di Fisica, Via P. Giuria 1,
10126 Torino, Italy 13]INFN Sezione di Roma Tor Vergata, Via della Ricerca
Scientifica 1, 00133 Roma, Italy 14]Clermont Université, Université Blaise
Pascal, CNRS/IN2P3, Laboratoire de Physique Corpusculaire, BP 10448, F-63000
Clermont-Ferrand, France 15]Université Claude Bernard Lyon I, 43 Bd du 11
Novembre 1918, 69100 Villeurbanne, France 16]CERN, Espl. des Particules 1,
1211 Meyrin, Svizzera 17]Università degli studi di Pavia, Corso Strada Nuova
65, 27100 Pavia, Italy 18]INFN Sezione di Napoli, Complesso Universitario di
Mone S. Angelo ed. 6, Via Cintia, 80126 Napoli, Italy 19]Massachusetts
Institute of Technology, 77 Massachusetts Ave, Cambridge, MA 02139, USA
20]Vrije Universiteit Brussel (VUB-ELEM), Dept. of Physics, Pleinlaan 2, 1050
Brussels, Belgium 21]Università degli studi di Roma Tor Vergata, Dipartimento
di Fisica, via della Ricerca Scientifica 1, 00133 Roma, Italy 22]Universidad
Iberoamericana, Dept. de Fisica y Matematicas, Mexico City 01210, Mexico.
23]Korea University, 145 Anam-ro, Seongbuk-gu, Seoul, South Korea
24]Université Paris-Saclay, 3 rue Joliot Curie, Bâtiment Breguet 91190 Gif-
sur-Yvette, France

# Preliminary results on the long term operation of RPCs with eco-friendly gas
mixtures under irradiation at the CERN Gamma Irradiation Facility

<EMAIL_ADDRESS><EMAIL_ADDRESS>M. Abbrescia G. Aielli R. Aly
M. C. Arena M. Barroso L. Benussi S. Bianco D. Boscherini F. Bordon A. Bruni
S. Buontempo M. Busato P. Camarri R. Cardarelli L. Congedo D. De Jesus Damiao
M. De Serio A. Di Ciacco L. Di Stante P. Dupieux J. Eysermans A. Ferretti G.
Galati M. Gagliardi S. Garetti R. Guida G. Iaselli B. Joly S.A. Juks K.S. Lee
B. Liberti D. Lucero Ramirez B. Mandelli S.P. Manen L. Massa A. Pastore E.
Pastori D. Piccolo L. Pizzimento A. Polini G. Proto G. Pugliese G. Rigoletti
A. Rocchi M. Romano A. Samalan P. Salvini R. Santonico G. Saviano M. Sessa S.
Simone L. Terlizzi M. Tytgat E. Vercellin M. Verzeroli N. Zaganidis [  [ [ [ [
[ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [

###### Abstract

Since 2019 a collaboration between researchers from various institutes and
experiments (i.e. ATLAS, CMS, ALICE, LHCb/SHiP and the CERN EP-DT group), has
been operating several RPCs with diverse electronics, gas gap thicknesses and
detector layouts at the CERN Gamma Irradiation Facility (GIF++). The studies
aim at assessing the performance of RPCs when filled with new eco-friendly gas
mixtures in avalanche mode and in view of evaluating possible ageing effects
after long high background irradiation periods, e.g. High-Luminosity LHC
phase. This challenging research is also part of a task of the European
AidaInnova project.

A promising eco-friendly gas identified for RPC operation is the
tetrafluoruropropene (C3H2F4, commercially known as HFO-1234ze) that has been
studied at the CERN GIF++ in combination with different percentages of CO2.
Between the end of 2021 and 2022 several beam tests have been carried out to
establish the performance of RPCs operated with such mixtures before starting
the irradiation campaign for the ageing study.

Results of these tests for different RPCs layouts and different gas mixtures,
under increasing background rates are presented here, together with the
preliminary outcome of the detector ageing tests.

###### keywords:

Gaseous detectors, Resistive-plate chambers, Eco-friendly gas mixtures, Beam
test, Aging studies

##  1 Introduction

The European Union has declared in EU regulation 517/2014 [1], the phase down
and limitation of fluorinated greenhouse gases (GHGs) production and usage.
Many scientific centers are pushing experimental collaborations to look for
possible eco-friendly replacements for the used gas mixtures. CERN, in
particular, is committed to reducing its GHG emissions and phase-down actions
have been put in place since 2020 [2]. Some studies report RPCs as the major
contributor to GHG emission from detector systems at the Large Hadron Collider
(LHC) during Run 1 and Run 2 [3].

The RPCs at CERN are mainly operated with mixtures composed of more than 90%
tetrafluoroethane (C2H2F4) and less than 1% of SF6, both GHGs characterized by
a global warming potential (GWP) of 1430 and 22800, respectively [1].

The search for eco-friendly alternatives plays a fundamental role in the
strategies to reduce GHGs emissions and possibly the relative operational
costs. In this context, the RPC ECOgas@GIF++ collaboration, which includes RPC
experts from the LHC experiments (CMS, ALICE, ATLAS, LHCb) and the gas group
of CERN, is pursuing the use of new gases in order to find a proper eco-
friendly gas mixture replacement. With this focus, the collaboration started
to study alternatives to C2H2F4. Possible candidates have been identified in
the family of the Hydro-Fluoro-Olefins (HFOs), due to the molecular similarity
with the C2H2F4 and low GWP ($\sim$6).

Within this gas family, tetrafluoropropene (C3H2F4), which comes in different
isomer forms [4], was of particular interest, mainly because two isomers, the
HFO-1234ze and HFO-123yf, are currently used as refrigerants in the industry,
making them rather available for purchase. The choice fell on HFO-1234ze due
to the moderate flammability of the yf isomer. In what follows, HFO-1234ze
will be referred to as HFO.

The HFO molecule contains the same number of Hydrogen and Fluorine atoms but
one more Carbon with respect to C2H2F4. On the other hand, the use of this gas
as tetrafluoroethane replacement leads to an increase of the detector working
voltage. This is not advisable since the currently installed detectors and
high voltage systems are not designed to operate at such high voltages, making
the full replacement of tetrafluoroethane with HFO alone not possible. A
mitigating solution was to add CO2 together with C3H2F4 to lower the operating
electric field by reducing the partial pressure of the gas mixture [5, 6, 7,
8].

The efforts of the RPC ECOgas@GIF++ collaboration are two-sided: on the one
hand various beam test campaigns have been carried out to fully characterize
RPC response when operated with various HFO/CO2-based gas mixtures. On the
other hand, the prolonged operation of RPCs, when flushed with eco-friendly
gas mixtures, is simulated by performing an aging test. During this kind of
studies, the detectors are exposed to an intense flux of $\gamma$ photons,
simulating the background conditions expected during the High-Luminosity (HL)
LHC phase [9] and the stability of their response is studied over time.

This article summarizes the main results obtained in the above-mentioned beam
test campaigns and aging studies. It is organized as follows: section 2
contains a brief description of the experimental setup, section 3 reports the
results obtained from the 2022 beam test campaign (results from the first RPC
ECOgas@GIF++ beam test campaign (in 2021) can be found in [10]), section 4
describes the results obtained so far from the aging campaign. Lastly, section
5 contains a summary of the results, providing also an outlook for possible
future developments of these studies.

##  2 Experimental setup

The studies of the RPC ECOgas@GIF++ collaboration are carried out at the CERN
Gamma Irradiation Facility (GIF++). This facility is equipped with a 12.5 TBq
${}^{\text{137}}$Cs source, which allows the users to perform long-term
irradiation studies (aging campaigns) with conditions similar to the ones
present during the HL-LHC phase [9]. The GIF++ is built on the H4 secondary
SPS beam line and it is traversed by a high-energy (150 GeV) muon beam,
produced by the interaction of the SPS proton beam on a number of fixed
targets [11].

The ${}^{\text{137}}$Cs source is equipped with an Aluminum angular correction
filter, used to transform the 1/$r^{\text{2}}$ ($r$ being the distance from
the source) dependence of the $\gamma$ flux to a uniformly distributed flux on
the xy plane (perpendicular to the beam), providing uniform irradiation on
large area detectors. The irradiation from the ${}^{\text{137}}$Cs source can
be modulated by means of a set of lead attenuation filters which are arranged
as a 3$\times$3 array, allowing for a total of 27 possible attenuation values,
from 1 (irradiator fully opened) to 46000 (maximum attenuation). The
irradiator can also be fully shielded (condition referred to as ”source-off”
in the following), allowing access to the GIF++ bunker. The peculiarity of the
facility is the possibility of combining the muon beam and irradiation with
photons, in order to study the detector response under different irradiation
conditions.

Each member of the RPC ECOgas@GIF++ collaboration provided one RPC prototype.
These have been installed on two mechanical supports, located inside the GIF++
and are characterized by different layouts (area, gas gap thickness, readout
system etc) as reported in table 1.

Table 1: Main features of the detectors from ECOgas@GIF++ collaboration |  |  | Gap/electrode |  |
---|---|---|---|---|---
Detector | Area (cm${}^{\text{2}}$) | Gaps | thickness | Strips | Readout
|  |  | (mm) |  |
ATLAS | 550 | 1 | 2/1.8 | 1 | Digitizer
LHCb/SHiP | 7000 | 1 | 1.6/1.6 | 64 | TDC
ALICE | 2500 | 1 | 2/2 | 32 | Digitizer
CMS RE11 |  1339.2+2298.5/4215.1111The CMS RE11 RPC gas gap layout consist in 2 gaps layout labelled as Top Narrow (TN) + Top Wide (TW) and Bottom (BOT). The gap areas are therefore expressed as TN+TW/BOT | 2 | 2/2 | 128 | TDC
BARI-1p0 | 7000 | 1 | 1/1.43 | 32 | TDC
EP-DT | 7000 | 1 | 2/2 | 7 | Digitizer
|  |  |  |  |

The gas system allows to create the desired gas mixture (by mixing up to four
gases) and sends it to the detectors, gas flow, relative humidity and mixture
composition are continuously monitored and a stop of the operations is issued,
by a dedicated software, in case of wrong mixture. The high voltage is
provided by means of a CAEN high voltage mainframe SY1527 [12], hosting two
high voltage boards (A1526N and A1526P [13])).

The applied high voltage is corrected for temperature and atmospheric pressure
variations, in order to maintain the effective high voltage
(HV${}_{\text{eff}}$) constant over time, according to the following equation,
used by the CMS collaboration [14]:

$HV_{app}=HV_{eff}\left[(1-\alpha)+\alpha\frac{P}{P_{0}}\frac{T_{0}}{T}\right]$
(1)

where P${}_{\text{0}}$ and T${}_{\text{0}}$ are reference values (293.15 K and
990 mbar) and $\alpha$ is an empirical parameter set to 0.8. Note that this
formula is slightly different from the one used in [10], describing the tests
performed in 2021. As a matter of fact, the formula used here was proved to
provide better stability of efficiency and other parameters over long periods
of time. It was not used for the 2021 data for organizational issues, since at
the time not all the groups from the RPC ECOGas@GIF++ collaboration were aware
and used it. Anyhow, the difference between HVapp computed with this or the
older formula is less than few ten of V, and this makes the results presented
here and the one in [10] directly comparable.

Both for aging and beam test studies, the data acquisition is carried out by
the WebDCS Ecogas [15], which is a web interface to a Detector Control System,
originally developed for the CMS collaboration studies at GIF++, and re-
adapted to the collaboration needs. This is a versatile system, which allows
the users to easily perform all the data taking, as well as produce on-the-fly
data quality monitoring plots.

During beam tests, a set of scintillators (coupled with photomultipliers) is
installed on the mechanical frames inside the bunker (internal scintillators)
and their coincidence with two external scintillators triggers the data
acquisition during the beam spill.

Figure 1 shows a sketch of the GIF++ bunker, with highlighted the positions of
the mechanical supports (black rectangles, at $\approx$3 and 6 m from the
source) and of the scintillators (blue rectangles the internal ones and red
rectangles the external ones). The total trigger area is equal to 10$\times$10
cm2. Lastly, the 137Cs source is also highlighted in the figure.

Figure 1: Sketch of the GIF++ bunker with the two mechanical supports
highlighted in black, internal scintillators in blue and external ones in red.
The black arrow represents the direction of the muon beam. The
${}^{\text{137}}$Cs source is also circled in green. The red lines show the
aperture of the irradiation field

##  3 Beam test results

This section reports the results obtained during the 2022 beam test campaign.
The first part of the section describes the detectors response when the 137Cs
source is fully shielded (source-off), whilst the second part shows the
behavior of the detectors when they are exposed to different values of photon
fluxes (source-on).

As it is reported in table 1, two different readout methods have been
employed: for some RPCs (ALICE, EP-DT and ATLAS), the readout strips have been
directly connected to a CAEN digitizer for full waveform studies (model DT5742
[16]1115 Gs/s 12-bit resolution for ALICE and ATLAS, and DT5730 [17]2221 Gs/s
and 14-bit resolution for EP-DT) while other detectors (SHiP, CMS RE11 and
BARI-1p0) use a specific front-end electronic board to discriminate the
signals which, once discriminated, are readout by means of a CAEN VME multi-
hit TDC (model V1190 [18]333128 channels and 100 ps time resolution on the
single hit).

Mixtures with different concentrations of CO2/HFO were tested, while the
fractions of i-C4H10 and SF6 were kept constant as to study the interplay
between CO2 and HFO. The mixtures are listed in table 2, together with their
GWP value, calculated as the average of the GWP of each component, weighted by
its mass concentration, according to what is prescribed in [1]. The GWP can be
used to compare the effects of the same mass of gas, expelled into the
atmosphere. However, the RPC gas systems at the LHC operate at a fixed rate of
gas volume changes; for this reason, to compare different mixtures (with
different specific masses), one can introduce the CO2-equivalent (CO2e) for 1
liter of gas mixture released into the atmosphere, expressed in grams per
liter. These values are reported in the last column of table 2. Note that the
first mixture in the table (STD) does not contain any HFO (or CO2): it
represents the standard gas mixture, currently employed in the ATLAS/CMS RPCs
and it has been taken as a reference to which the eco-friendly alternatives
are compared.

Table 2: Composition of the gas mixtures used in the tests described in this paper Mixture | C2H2F4 % | HFO % | CO2 % | i-C4H10 % | SF6 % | GWP | CO2e (g/l)
---|---|---|---|---|---|---|---
STD | 95.2 | 0 | 0 | 4.5 | 0.3 | 1485 | 6824
MIX0 | 0 | 0 | 95 | 4 | 1 | 730 | 1480
MIX1 | 0 | 10 | 85 | 4 | 1 | 640 | 1490
MIX2 | 0 | 20 | 75 | 4 | 1 | 560 | 1495
MIX3 or ECO3 | 0 | 25 | 69 | 5 | 1 | 527 | 1519
MIX4 | 0 | 30 | 65 | 4 | 1 | 503 | 1497
MIX5 or ECO2 | 0 | 35 | 60 | 4 | 1 | 476 | 1522
MIX6 | 0 | 40 | 55 | 4 | 1 | 457 | 1500
|  |  |  |  |  |  |

It is worth noting that the CO2e for all the mixtures tested is quite similar;
this is due to the fact that the SF6 concentration is the same and it is the
only GHG in the mixtures tested and that they are $\approx$4/4.5 times lower
with respect to the standard gas mixture one.

###  3.1 Results without gamma background

This section summarizes the main outcomes that have been obtained during the
2022 test beam campaigns, carried out by the RPC ECOgas@GIF++ collaboration.
In particular, the results obtained when the 137Cs source is fully shielded
and no background radiation is present on the detectors (source-off) are
reported here.

####  3.1.1 Source off Efficiency and Working Point

Both the digitizers and the TDCs provide a timestamp for each hit they
register, regardless of its origin. This time information can be used to
separate the muon-induced hits from those coming from other sources, such as
noise and/or the $\gamma$ background. Figure 2 shows the time profiles
(distribution of the hits arrival times for a fixed high voltage value)
obtained for the ALICE RPC, operated with the standard gas mixture at 90%
efficiency at source-off (left panel) and under the highest possible
(irradiatior fully opened) $\gamma$ background condition (right panel).

Figure 2: Hit time profile obtained with the ALICE RPC when flushed with the
STD gas mixture at 90% efficiency. Left panel: source-off condition. Right
panel: highest possible (irradiatior fully opened) $\gamma$ background
condition

In the case of the TDCs, the width of the acquisition window was set to 5000
ns while in the case of the digitizer to: 500 ns for the ATLAS detector, and
1040 ns and 1024 ns for the EP-DT and ALICE chambers, respectively. The choice
of TDC or digitizer was made due to different requirements of the groups
involved (i.e. the digitizer is used to study the analog response of the
detector while the TDCs, coupled with front-end electronics, are used to
simulate real-life conditions for RPCs in the LHC). The peaks that are clearly
visible in both panels (with a width of $\approx$25 ns) of figure 2 correspond
to an accumulation of muon-induced events (since the time interval between the
trigger and the muon hit is the same for each event) while the others,
uniformly distributed, are due to the noise/gamma-induced hits.

The chamber efficiency can then be computed as the ratio between the number of
events whose time falls inside the muon window and the number of triggers.

The left and right panels in figure 3 show the efficiency and absorbed current
density (reported in nA/cm2, to compare it across detectors with different
active areas) as a function of the effective high voltage applied to the
detectors without $\gamma$ background for three different mixtures: STD, ECO2
(30/65 HFO/CO2) and ECO3 (25/69 HFO/CO2) for the ATLAS (2 mm single gas gap)
and the BARI-1p0 (1 mm single gas gap) respectively.

(a)

(b)

Figure 3: Efficiency and absorbed current density as a function of HVeff,
without $\gamma$ background. Left panel: ATLAS RPC. Right panel: BARI-1p0 RPC

The efficiency data points were interpolated using the logistic function
reported in equation 2:

$\varepsilon(HV)=\frac{\varepsilon_{max}}{1+e^{-\beta(HV-HV_{50})}}$ (2)

where the free parameters are: $\varepsilon_{max}$, which represents the
asymptotic maximum efficiency (plateau efficiency); $\beta$, which is related
to the steepness of the efficiency curve, and HV50, which represents the
voltage where the efficiency reaches 50% of its maximum. These values are used
to compute the voltage corresponding to the Working Point (WP), according to
the definition used by the CMS collaboration [19]:

$\text{WP}=\frac{\log{19}}{\beta}+HV_{50}+\text{150 V}$ (3)

Table 3 reports the most significant parameters for the detectors shown in
figure 3. It is possible to observe that, in the case of the ATLAS RPC, the
plateau efficiency is compatible for the different mixtures while, in the case
of a thinner gap, such as the BARI-1p0 detector, the efficiency greatly
decreases in the case of the eco-friendly alternatives. A possible explanation
could be that the CO2 produces a smaller number of primary ion-electron
clusters due its lower density [20], leading to a more significant efficiency
drop in thinner gas gaps. This is further confirmed by observing that, moving
from ECO3 to ECO2, the CO2 concentration decreases (and the HFO concentration
increases), making the mixture denser, and the plateau efficiency increases.
Furthermore, as anticipated in section 1, the inclusion of HFO to the mixture
tends to move the detector WP to higher values. The increase with respect to
the standard gas mixture is similar for both detectors for ECO3 (around
0.3/0.4 kV) while for ECO2 it is around 1 kV for the 2 mm gas gap detector and
around 0.8 kV for the 1 mm gas gap RPC.

Table 3: Source off WP and plateau efficiency for the ATLAS and the BARI-1p0 RPCs Mixture | Detector | WP [V] |  $\varepsilon_{max}$ [%]
---|---|---|---
STD | ATLAS | 9925.7 | 96.71
ECO2 | ATLAS | 11021.9 | 95.92
ECO3 | ATLAS | 10200.7 | 94.56
STD | BARI-1p0 | 5903 | 93.89
ECO2 | BARI-1p0 | 6646.2 | 89.73
ECO3 | BARI-1p0 | 6301.2 | 88.38
|  |  |

####  3.1.2 Signal charge distribution and large signals contamination

By using a digitizer, one has access to the waveform of each signal detected
by the RPCs under test and this allows to perform a full characterization of
the detector response, especially in terms of signal charge and contamination
from large signals. The starting point in the signal charge calculation is the
discrimination between signals and noise. In the following, results from EP-DT
and ALICE detectors are presented, hence a few details on the procedure
employed in the analysis are described here. In the case of the EP-DT
detector, a waveform is considered to contain a signal from a muon if its
amplitude is above 2 mV with respect to the baseline while in the case of
ALICE, the threshold was set to five times the RMS of the signal in a region
where no muon signal is expected (noise window).

The signal charge is then calculated by integrating the signals passing the
above selection criterion in a suitable integration window. In particular, the
range for signal integration is determined as follows (a visual reference is
also reported in figure 4):

  * •

The first and last samples where the signal is above threshold are determined
for each strip

  * •

Starting from the first (last) point the signal is swept forward (back) and
the discrete derivative between two consecutive samples is calculated. When
this changes sign, it means that there is a change in the signal slope and the
last point before the sign change is assumed to be the start (end) point of
the integration interval (for a complete description of the algorithm the
reader can refer to [21, 22])

  * •

The charge calculated for each strip is then summed, to get the total charge
per event

Figure 4 shows an example of a signal as seen by the ALICE RPC, when flushed
with the standard gas mixture: the horizontal blue line represents the
threshold in the specific event and the two black markers show the start and
end of the integration interval just described. Since the signal is readout on
a 50 $\Omega$ resistor, to find the value of signal charge, the result of the
signal integration is divided by 50 $\Omega$.

Figure 4: Example of muon signal recorded by the digitizer of the ALICE RPC.
The black circles refer to the start/end of the integration interval for
signal charge calculation and the blue line represent the threshold

The left and right panels of figure 5 show, respectively, the charge
distributions for the EP-DT and ALICE detector when the applied high voltage
is the closest to the estimated working point for the tested gas mixtures.
Note that the average threshold used for the ALICE detectors ($\approx$1.6 mV)
is lower with respect to the EP-DT one (2 mV), and this likely affects the
average values of the signal charges measured in the two cases. Indeed, it
appears that for the ALICE detector, the average charge values are slightly
lower, with respect to the EP-DT one.

(a)

(b)

Figure 5: Signal charge distributions at the calculated working point without
$\gamma$ irradiation. Left panel: EP-DT RPC. Right panel: ALICE RPC. Note that
different gas mixtures are used in the two detectors

In both cases, the black distribution in figure 5 refers to the standard gas
mixture, while the others refer to all the tested eco-friendly alternatives.
For all mixtures, two peaks can be observed, the one at lower charge values
coming from the avalanche contribution while the one at higher charge values
from larger signals. Usually these are referred to as streamers [23] but, in
the case of the eco-friendly alternatives, not all signals in the right peak
of the distribution in figure 5 satisfy all the criteria to be defined
streamers (i.e. they are not always accompanied by a precursor signal and they
might be characterized by multiple delayed peaks). For this reason, these will
be referred to as ”large signals” from here on. In general, for the eco-
friendly alternatives, the avalanche peak is shifted towards higher values
with respect to the standard gas mixture (this observation is consistent with
the higher absorbed current, as reported in 3.1.1 and figures therein).
Moreover, the fraction of large signals is generally larger, although this
value seems to be decreasing for increasing HFO concentration in the mixtures.
Of course, the decrease of these kind of signals comes with the price of a
higher working point, mainly due to the quenching effect of adding more HFO to
the mixture.

In order to quantify this value, we tag as ”large signals” all events
characterized by a charge larger than 16 pC (this value was chosen by
observing that the two peaks in the charge distributions are separated at
$\approx$ the 16 pC mark). The large signal probability can then be defined as
the ratio between the number of these signals and the total number of events.
Figure 6 shows, in the left panel, the values of large-signal probability at
the working point, for the EP-DT detector, while the right panel shows the
large-signal probability as a function of the applied high voltage, in the
case of the ALICE detector. It is possible to observe how, for increasing HFO
concentration, the contamination from large signals at WP reaches similar
values as the standard gas mixture but it tends to increase more sharply for
voltages above the WP. This leads to a narrower range of applicable high
voltage which grants both a high detection efficiency ($>$ 95%) as well as a
low large signal probability ($<$ 5%). The large signals contamination for the
ALICE detector is reported as a function of the high voltage minus working
point; in this way, the point at 0 V corresponds to the WP for all mixtures.

(a)

(b)

Figure 6: Left panel: Large signal probability at WP for the EP-DT detector.
Right panel: large signal probability as a function of HVeff for the ALICE
detector

####  3.1.3 Source off summary

Using the previously shown results, a few conclusions can be drawn. First of
all, for thinner gas gap detectors (such ad the BARI-1p0), an HFO fraction
above 50% is advisable, in order to reach a high enough efficiency plateau.
This is less true for the 2 mm detectors (as in the case of the ATLAS RPC),
where mixtures with 25% HFO already provide an efficiency $>$95% at working
point. The detector working point also increases if the HFO concentration
increases, at a level of $\approx$1 kV for every 10% HFO added to the mixture.
Lastly, by studying the signal charge and large signals contamination, it is
worth noting that the average signal charge is higher for all the eco-friendly
alternatives, leading to the higher absorbed currents observed. Moreover, for
increasing HFO fractions, the average avalanche charge and large signal
contamination both decrease.

###  3.2 Results with gamma background

The performance of the chambers under test were also investigated using
different combinations of attenuation filters to shield the 137Cs source.
Figure 7 shows the efficiency and absorbed current density, as a function of
HV${{}_{e}ff}$, for the ATLAS detector (located at $\sim$3 m from the source)
with the STD gas mixture (left panel) and two HFO-based candidates, ECO2
(middle panel) and ECO3 (right panel), in different conditions of $\gamma$
background.

(a)

(b)

(c)

Figure 7: Efficiency and current densities for the ATLAS detector in different
regimes of irradiation. Left panel: STD mixture. Central panel: ECO2 mixture.
Right panel: ECO3 mixture

The efficiency curves with the three mixtures are shifted at higher voltages
for increasing $\gamma$ rates. This phenomenon occurs because of the increased
$\gamma$ background, which leads to a higher absorbed current. Flowing through
the resistive chamber electrodes, this current leads to a voltage drop across
them which, in turns, leads to a reduction of the electric field inside the
gap, which must be recovered by increasing the supplied high voltage. The
right panel of figure 7 shows that for the ECO3 gas mixture, the efficiency
decrease due to the $\gamma$ background is the highest among the three. This
happens because the CO2 content in this mixture is the highest. On the other
hand, the current densities follow the expected behavior, increasing with the
$\gamma$ background rate. The highest increase is observed if the CO2
concentration increases (ad is the case for ECO3).

####  3.2.1 Gamma cluster rate

The $\gamma$ cluster rate was measured for all the detectors with different
absorption factors. This approach allows to assess the rate capabilities of
each RPC at specific distances from the source. The $\gamma$ cluster rate is
calculated using the data collected when no beam is present. The RPC response
is sampled using a random trigger (a pulse sent to the DAQ modules with a
given frequency) and, for each trigger, the data is grouped in clusters (i.e.
a $\gamma$ can lead to an above-threshold signal on more than one adjacent
strip). The number of $\gamma$ clusters is then counted and divided by the
total acquisition time (5000 ns $\times$ number of random triggers) multiplied
by the detector active area (to get a measure in Hz/cm2). Figure 8 shows the
$\gamma$ cluster rates for the ALICE and EP-DT detectors (both featuring a gas
gap thickness of 2 mm and located at 6 and 3 m from the source respectively)
for all the tested gas mixtures. The results show similar rates measured by
both detectors. The ALICE detector registered lower rates compared to those
observed with the EP-DT detector (for the same value of absorption factor), in
agreement with the respective distance from the $\gamma$ source. The $\gamma$
rates measured for STD, ECO2, and ECO3 gas mixtures, with BARI-1p0 detector
(located at $\approx$3 m from the source) are shown in figure 9. The results
in this case are comparable with the ones measured by EP-DT detector.

(a)

(b)

Figure 8: Gamma cluster rate at working point as a function of the attenuation
filter configuration (ABS). Left panel: ALICE RPC. Right panel: EP-DT RPC
Figure 9: $\gamma$ cluster rate at working point for BARI-1p0 RPC as a
function of the attenuation filter configuration (ABS)

####  3.2.2 Efficiency and Working Point

The efficiency under irradiation was measured following the method described
in section 3.1.1. Figure 10 shows the values of plateau efficiency
($\epsilon_{max}$ as a function of the measured $\gamma$ cluster rate for the
ALICE and EP-DT detectors operated with several candidate mixtures.

(a)

(b)

Figure 10: Plateau efficiency ($\epsilon_{max}$) as a function of the $\gamma$
cluster rate measured at WP. Left panel: ALICE detector. Right panel: EP-DT
detector

The same trend already shown for the ATLAS detector in figure 7 is also
visible in this case (i.e. the plateau efficiency decreases if the irradiation
increases). The highest efficiencies (and the smallest decrease for increasing
background) for both detectors are obtained using the STD mixtures, while with
the eco-friendly candidates the efficiency decrease at higher rates is more
pronounced with respect to the STD mixture; a similar behavior is observed for
the ALICE and EP-DT detectors.

The plateau efficiency measured with the BARI-1p0 detector at different
$\gamma$ cluster rates is shown in figure 11. Despite the fact that the
highest efficiency is reached with the STD mixture, the efficiency results in
the range 90-79% up to 800 Hz/cm2 with the ECO2. Moreover, it is remarkable
that in this case the plateau efficiency measured with the ECO3 gas mixture
decreases more rapidly with respect to ECO2 and STD.

Figure 11: Plateau efficiency ($\epsilon_{max}$) for the BARI-1p0 RPC as a
function of the $\gamma$ cluster rate measured at WP

The working points were calculated by fitting the efficiency curves for each
set of attenuation filters by using equation 2. Figures 12 and 13 show the
working points for the different background rates measured with the ALICE,
CERN EP-DT and BARI-1p0 RPCs. The ALICE and EP-DT chambers are characterized
by similar working points and the value is shifted following the amount of CO2
present in the mixtures. On the other hand, figure 13 shows that the WP, for
the BARI-1p0 RPC, characterized by a 1 mm gas gap, shifts of just few hundred
Volts because of the use of an eco-friendly gas mixture. The values, in this
case, are shifted from 6.65 to 6.9 kV for ECO2 and from 6.3 to 6.45 kV for
ECO3, smaller shifts than for the detectors characterized by a 2 mm gas gap.
The working point shift between mixtures is lower with respect to the ones
calculated for ALICE and EP-DT RPCs due to the thinner gap used in BARI-1p0.

(a)

(b)

Figure 12: WP as a function of the $\gamma$ cluster rate measured at WP. Left
panel: ALICE RPC. Right panel: EP-DT RPC Figure 13: WP for the BARI-1p0
detector as a function of the $\gamma$ cluster rate measured at WP

####  3.2.3 Muon cluster size

The muon cluster size quantifies the number of neighboring strips fired due to
an avalanche produced by a muon. This value is a significant feature because
of its direct impact on the detector spatial resolution. In the following
figures, the cluster size is expressed in cm in order to enable the comparison
between the different detectors with different strip pitch. Figure 14 shows
the muon cluster size values at WP as a function of the $\gamma$ cluster rate
for the ALICE and BARI-1p0 detectors. In both RPCs, the values are slightly
higher for the eco-friendly candidates at low $\gamma$ rates while at higher
rates, the difference between the cluster sizes becomes less important.
Moreover, one can observe that for all the tested gas mixtures, a decreasing
trend with increasing irradiation is observed.

(a)

(b)

Figure 14: Muon cluster size at WP as a function of $\gamma$ cluster rate
measured at WP. Left panel: ALICE RPC. Right panel: BARI-1p0 RPC

##  4 Preliminary aging studies

This section describes some preliminary results obtained from an aging test,
covering the period from July 2022 to July 2023. First, a brief description of
the general methodology used in the data-taking is provided; following this, a
summary of the main results is presented.

###  4.1 Methodology

During the aging test, the detectors are flushed with the selected gas
mixture, the high voltage is set to a fixed value and the stability of the
absorbed current (measured by the high voltage power supply, with a precision
of 0.1 $\mu$A) is monitored over time. The webdcs applies the correction for
temperature/pressure changes, according to equation 1, as explained in section
1, in order to maintain a constant HVeff on the detectors.

The values of current, applied and effective high voltage are saved every 30
seconds for data analysis. Moreover, once a week, the 137Cs source is fully
shielded (source-off) and a measurement of the absorbed current without
$\gamma$ irradiation is performed. This current is familiarly called ”dark
current” and it is an important parameter to monitor throughout the aging
test, since its increase could be a sign of detector aging.

In order to numerically quantify the progress of the aging, we use the
integrated charge density, defined as the integral over time of the current
density passing through the detector and it is measured in mC/cm2. By looking
at the left panel of figure 15, one can see an example of the absorbed dark
current density as a function of HVeff (I(HV) curve) for the EP-DT detector
when flushed with the ECO2 gas mixture; it is possible to observe that, even
for voltages well below the threshold for avalanche multiplication processes
(7/8 kV for a 2 mm gap, as is the case for EP-DT), a non-zero current is
flowing. This is the Ohmic component of the dark current and it is, in
principle, not flowing through the gas but rather through some other
conductive paths in the detector. Most likely, this Ohmic component is not
relevant for aging processes, since it is not related to discharge processes
happening in the gas, which may lead to the dissociation of HFO and the
production of harmful pollutants. Source-off current density vs HV curves are
measured weekly and this allows one to also monitor the Ohmic component of the
dark current.

Since the irradiation is carried out at fixed HVeff (10.6 kV for the 2 mm gaps
and 8.8-9 kV for the 1.6 mm one, in the case of the ECO2 gas mixture, as it
will be better explained in section 4.2), it is useful to estimate the Ohmic
part of the dark current at said voltage. To this aim, a linear interpolation
of the current density vs HVeff curve is carried out between 0 and 5 kV, for
the 2 mm gaps and between 0 and 4 kV for the 1.6 mm one. The straight line is
then extrapolated to the irradiation voltage, providing the required
estimation of the Ohmic dark current density. This is then subtracted from the
total current density measured, obtaining the current density flowing through
the gas and related to the $\gamma$ irradiation. Since a weekly dark current
measurement is performed, one can subtract its Ohmic component from the
measured current density for the whole irradiation period (i.e. for each
irradiation period we subtract its closest (in time) estimation of Ohmic dark
current). An example of this procedure is shown in the right panel of figure
15, where the current density absorbed under irradiation is shown for the EP-
DT detector. The blue markers represent the current density measured by the
high voltage module, the ones in red show the current when the Ohmic component
was subtracted. The current density to which its Ohmic component was
subtracted is used to compute the integrated charge density.

(a)  (b)

Figure 15: Left panel: example of the dark current density as a function of
HVeff for the EP-DT detector. The Ohmic part of the dark current is clearly
visible as well as the linear interpolation to obtain the Ohmic dark current
at the irradiation voltage (represented by the intersection between the
vertical blue line and the red straight line). Right panel: example of current
density under irradiation with and without the Ohmic part of the dark current
(EP-DT detector). The discrete step observed towards the end of the period
corresponds to a change in the irradiation conditions, which causes to a
current reduction

###  4.2 Main results from aging studies

During the irradiation studies carried out between July 2022 and July 2023,
the detectors have been flushed with the ECO2 gas mixture and, for the most of
these studies, the source attenuation filter was set to a value of 2.2. As
anticipated, the HVeff chosen for the irradiation corresponds to 10.6 kV (for
the 2 mm gas gap detectors) and 8.8/9 kV (for the 1.6 mm SHiP RPC, detector
fully characterized with beam whose results have not been shown in this paper
for the sake of avoiding repetition). Note that the BARI-1p0 detector was not
included in the aging studies since it experienced a large current increase
following the beam test campaigns and it was removed from the setup. With this
HVeff the detectors are not fully efficient. The reason why this voltage was
chosen is to limit the currents absorbed by the detectors (indeed, by looking
at figure 7 the current absorbed with a background of 500 Hz/cm2 with ECO2 is
$\approx$twice as much as with respect to the standard gas mixture). Indeed,
in the LHC experiments the detectors are not kept with high voltage on at all
times, hence aging studies with too large currents would not be representative
of real life conditions, possibly leading to the appearance of artifacts which
would not be observed when operating the detectors.

Figure 16 shows the absorbed current density (with and without the Ohmic
component of the dark current, in blue and green respectively) as well as the
HVeff, as a function of the integrated charge density (this choice was made
since the irradiation campaign is sometimes interrupted and this would leave
some gaps in the chart) for the CMS RE11 TN gap (in the left panel) and SHiP
chamber (in the right panel).

(a)

(b)

Figure 16: Absorbed current density (with and without the Ohmic dark current)
and HVeff during an $\approx$ one year exposure to the GIF++ 137Cs source as a
function of the integrated charge density. Left panel: CMS RE11 TN gap. Right
panel: SHiP detector

From the figure, one can see that HVeff is constant in time, as expected. The
clear changes in the absorbed current density, that are visible in the figure,
correspond to changes in the configuration of the other experimental setups
placed in front of the detectors (indeed if other setups are inserted/removed
the radiation shielding changes and so does the absorbed current). The
portions of the charts where HVeff is changing (e.g. around the 10 mC/cm2
mark) correspond to the measurement of I vs HVeff, while the regions where the
current is smaller correspond to periods where either the source was off or
the absorption factor was set to values higher than 2.2.

In the case of the SHiP detector (right panel of figure 16), the HVeff was
increased in steps from 8.8 kV, corresponding to 50% efficiency, up to 9.8 kV,
corresponding to plateau efficiency, in order to study the evolution of the
current accordingly. It is possible to see that, towards $\approx$80 mC/cm2,
the absorbed current starts to fluctuate in a more pronounced way. To
investigate this effect, the HVeff of the SHiP RPC was reduced and the current
absorbed with this lower value is being closely monitored. Indeed, as reported
in [24], keeping the detectors with a lower than nominal applied high voltage
has been proven to somewhat reduce the current drawn by the detectors (a
possible explanation for this might be the burn of small imperfections of the
bakelite).

It is also useful to monitor the evolution of the dark current (both its Ohmic
as well as the total components). Indeed, as anticipated, an increase of
absorbed dark current could be a sign of potential detector aging. Figure 17
summarizes this trend, by showing both components of the dark current at the
irradiation voltage (10.6 kV for the 2 mm detectors and between 8.8 and 9.8 kV
for the 1.6 mm gap), as a function of the integrated charge density for the
CMS RE11 TN gap (left panel) and the SHiP detector (right panel).

(a)

(b)

Figure 17: Dark current density (total and extrapolated Ohmic part) at the
irradiation voltage (10.6 and 8.8 up to 9.8 kV for the 2 and 1.6 mm gaps
respectively) as a function of the integrated charge density following an
exposure of around one year to the GIF++ 137Cs source. Left panel: CMS RE11 TN
gap. Right panel: SHiP detector (the different HVeffvalues are also reported
on the chart for this detector)

In figure 17, one can see that, sometimes, the same current density value is
reported for different integrated charge densities. This is due to the fact
that a single source off current scan is performed per week but multiple
irradiation scans could be started during the same week so the same value of
dark current density is used for multiple irradiation scans. The discrete step
that can be seen in the right panel of figure 17 at $\approx$40 mC/cm2
corresponds to the fact that the irradiation voltage for the SHiP detector was
increased, hence the higher values. The last few points of the same chart
refer to the abnormal increase in absorbed current density already reported in
figure 16.

Also, from figure 17, it is possible to see that the Ohmic component of the
dark current shows an increasing trend at the start of the irradiation, while
it reaches a more stable behavior for higher values of integrated charge
density. For what concerns the total dark current density, it shows a more
uniform increasing trend during the whole irradiation campaign. This needs to
be closely monitored in time, especially for some specific detectors, which
showed a more significant increase of the total dark current density than
others. To shed some light on this, the RPC ECOgas@GIF++ collaboration is
planning to start the monitoring of other parameters, such as the presence of
possible current leaks on the mechanical frame and the production of
fluorinated impurities in the exiting gas mixture. Moreover, one also needs to
monitor the detectors performance in terms of response to cosmic/beam muons,
with time. This has been done in July 2023, when another beam test campaign
was carried out and the data gathered is currently being analyzed. In this
way, one will be able to estimate the performance evolution and have a first
insight on the real aging observed on the detectors.

The integrated charge density during around one year of exposure to the GIF++
137Cs source, is shown in figure 18. The left panel shows the results for the
three gaps of the CMS RE11 RPC while the right panel refers to the other
detectors. The fact that the integrated charge density is not exactly the same
across the detectors, can be explained by considering that the irradiation
voltage chosen does not exactly correspond to the same efficiency value.

(a)

(b)

Figure 18: Accumulated charge density as a function of time for all the RPC
ECOgas@GIF++ collaboration detectors. Left panel: CMS RE11 RPC (three gaps).
Right panel: ALICE, ATLAS, EP-DT and SHiP RPCs

##  5 Conclusions

This paper explored some of the most recent activities of the RPC ECOgas@GIF++
collaboration. These have been focused on performance and aging studies on RPC
detectors operated with different eco-friendly gas mixtures, where C2H2F4 has
been replaced using mixtures with various concentrations of HFO and CO2.

During the beam tests, it was observed that the plateau efficiency reached
without irradiation increases at increasing HFO concentrations and so does the
detector WP. It was observed that this shift is around 1 kV for every 10% HFO
added to the gas mixture.

The average value of signal charge, for all the HFO-based gas mixtures, is
generally larger, with respect to the standard gas mixture and also a higher
fraction of events with large charge content is observed. Both values tends to
decrease if the HFO concentration increases; reaching, at the detector WP,
similar values to the standard gas mixture. It was nevertheless observed that
the useful operating region (i.e. the high voltage range where the efficiency
is above 95% and the large signal contamination is below 5%) is reduced for
the eco-friendly alternatives (since the number of events with large signals
increases more sharply with the voltage, if compared to the standard gas
mixture).

For what concerns the RPC response under $\gamma$ irradiation, different
observations can be made: first of all, the efficiency curves shift to higher
voltages (the same can be said also for the detector WP) if the background
level increases; secondly, the plateau efficiency decreases. These effects can
be partly explained by considering that when the detectors are exposed to an
intense $\gamma$ background, the absorbed current increases and, circulating
through the resistive bakelite electrodes, this leads to a voltage drop across
the electrodes themselves, leading to a reduction of the voltage applied to
the gas, leading to a lower gain and lower efficiency. It was observed that
the maximum efficiency reduction is $\approx$1$\div$2 % (between source off
and the highest irradiation condition) for the standard gas mixture while it
ranges from 8 down to$\div$4 % for the HFO-based gas mixtures (the effect is
less visible if more HFO is added to the mixture).

For what concerns the preliminary results obtained from the aging studies, an
irradiation campaign with the ECO2 (35/50 HFO/CO2) was started in July 2022.
The stability of the absorbed current (both with and without irradiation) was
monitored for around one year now. It was observed that the current under
irradiation is quite stable over time. The Ohmic component of the dark current
also appears quite stable over time while the trend of the total dark current
is more subject to fluctuations. These effects are under investigation at the
moment but what will of the utmost importance in the future is a continuous
monitor of all the detector performance (i.e. efficiency, prompt charge, pulse
spectrum etc.). A beam test campaign has been carried out in July 2023, to
perform a first comparison with the previous data; the analysis is still
ongoing and, soon, this comparison will be made.

All in all, the efforts of the RPC ECOgas@GIF++ collaboration have lead to
some breakthrough in the search for eco-friendly alternative gas mixtures. The
ongoing aging campaign, complemented by periodic beam test studies, will help
to shed some light on the long-term behavior of RPC detectors operated with
eco-friendly alternatives.

## Declarations

##### Funding

This work is part of the AIDA Innova project, which received funding from the
European Union’s Horizon 2020 Research and Innovation programme under GA n.
101004761. In particular, this work is part of the Working Package 7.2
(Multigap RPCs (MRPCs) for fast timing and Eco-friendly gas mixtures for RPCs)

##### Data availability statement

The data shown in this article can be made available by the corresponding
author(s) upon request

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# The nematic susceptibility of the ferroquadrupolar metal TmAg2 measured via
the elastocaloric effect

Elliott W. Rosenberg  1) Geballe Laboratory for Advanced Materials and
Department of Applied Physics, Stanford University, Stanford, CA 94305, USA
Matthias Ikeda  1) Geballe Laboratory for Advanced Materials and Department of
Applied Physics, Stanford University, Stanford, CA 94305, USA  Ian R. Fisher
1) Geballe Laboratory for Advanced Materials and Department of Applied
Physics, Stanford University, Stanford, CA 94305, USA

###### Abstract

Elastocaloric measurements of the ferroquadrupolar/nematic rare-earth
intermetallic TmAg2 are presented. TmAg2 undergoes a cooperative Jahn-Teller-
like ferroquadrupolar phase transition at 5K, in which the Tm3+ ion’s local
$4f$ electronic ground state doublet spontaneously splits and develops an
electric quadrupole moment which breaks the rotational symmetry of the
tetragonal lattice. The elastocaloric effect, which is the temperature change
in the sample induced by adiabatic strains the sample experiences, is
sensitive to quadrupolar fluctuations in the paranematic phase which couple to
the induced strain. We show that elastocaloric measurements of this material
reveal a Curie-Weiss like nematic susceptibility with a Weiss temperature of
$\approx 2.7K$, in agreement with previous elastic constant measurements.
Furthermore, we establish that a magnetic field along the c-axis acts as an
effective transverse field for the quadrupole moments.

##  I Introduction

Intermetallic rare-earth compounds, in which the partially filled $4f$
orbitals associated with the rare-earth ions form a lattice of local
electronic degrees of freedom, can provide models for studying cooperative
phase transitions arising from microscopically well-understood multipole
interactions [1]. If the $4f$ electronic orbitals form a degenerate ground
state, then the system (at low temperatures relative to any excited states)
can be approximated as an sublattice of electronic multipolar moments coupled
to non-$4f$ itinerant metallic degrees of freedom [2]. Materials such as
PrPb3, DySb, and YbRu2Ge2 all exhibit low temperature phase transitions in
which the $4f$ sublattice plays a crucial role in driving multipolar phase
transitions [3, 4, 5, 6, 7]. TmAg2 is also such a metal, in which the $4f$
ground state $\Gamma_{5}$ doublet spontaneously splits through undergoing
$B_{1g}$ (i.e. $x^{2}-y^{2}$ symmetry) ferroquadrupolar order at 5K, breaking
the $C_{4}$ rotational symmetry of the tetragonal crystal structure [8, 9].
This phase transition does not break translational symmetry, and so from a
symmetry perspective is equivalent to the electronic nematic phase transitions
that have been recently investigated in a variety of material families
including the iron pnictide and chalcogenide superconductors [10, 11, 12, 13].

The effective Hamiltonian describing the low-energy properties of $4f$ ions
incorporated in crystalline solids is understood in great detail. For cases in
which the Kondo energy scale is vanishingly small [1], the $4f$ electronic
wavefunctions are spatially localized, and so form a lattice of local
electronic sites. Each site has strong spin-orbit coupling which yields an
electronic multiplet that is described by a total angular momentum number J.
Due to the local nature of the multiplet, the crystal electric field (CEF)
from the surrounding ligands acts as a weak perturbation, splitting the
degeneracy of the spherical harmonic basis states. Coupling between $4f$
orbitals of different lattice sites arises both from the interaction of the
quadrupolar degrees of freedom with the lattice, and the generalization of the
Ruderman–Kittel–Kasuya–Yosida (RKKY) interaction, mediated by the conduction
electrons in metals [1]. This provides an effective exchange interaction
between the $4f$ sites, analogous to the exchange interaction that appears in
magnetic Ising models.

Significantly, the low-energy states of these $4f$ electronic multiplets can
exhibit degeneracies, the multiplicity of which is determined by the point-
group symmetry and charge distribution surrounding the $4f$ ion. For
tetragonal materials with even angular momentum number J, $\Gamma_{5}$
doublets, (i.e. $E_{g}$ symmetry, which transforms like ($xz$, $yz$)), can be
the ground state of the $4f$ system, as is the case for TmAg2. At low enough
temperatures (relative to the higher energy states determined by the
crystalline effective field), thermal occupation of the excited $4f$ states
can be neglected, and so materials with this ground state configuration are
ideal candidates to compare to a pseudo-spin 1/2 Ising model in which sites
are coupled via both the lattice and the metallic degrees of freedom [2].

As shown in previous work (Ref. [2]), in this restricted Hilbert space for
TmAg2, $B_{2g}$ strain (i.e. $\varepsilon_{xy}$) and magnetic field $H_{z}$
directed along the c-axis map precisely to transverse fields, and so this
system maps to a transverse field Ising model with an additional coupling
between the sites and the lattice [14]. Notably, a magnetic field has been
shown to suppress this phase transition to 2K for an applied field of 3T,
allowing exploration of a possible quantum phase transition if the phase
diagram could be followed to lower temperatures [8]. However, as mentioned
previously, it should be emphasized that TmAg2 contains metallic degrees of
freedom that interact with the local $4f$ Ising lattice, unlike the
paradigmatic case of $4f$ ferroquadrupolar order in the insulating compound
TmVO4 [15, 14, 16]. We might still expect to tune the ferroquadrupolar
transition towards a metallic nematic quantum critical point via the
application of a transverse field, but this assumes the metallic degrees of
freedom don’t couple strongly enough to the $4f$ sublattice to fundamentally
change the low-temperature behavior of the total coupled system. The
interactions between the metallic part of this material and the $4f$
sublattice are then of particular interest in the proximity of the potential
quantum critical point.

TmAg2 undergoes only one phase transition, a ferroquadrupolar phase transition
occuring at 5K [8]. The local $4f$ electronic multiplets, which for the Tm3+
ion harbor quadrupolar degrees of freedom, develop spontaneous electronic
quadrupolar moments which collectively align at this phase transition. The
macroscopic quadrupolar order parameter breaks the four-fold rotational
symmetry of the lattice, and so the lattice distorts (i.e develops a
spontaneous strain) in a similar fashion due to its coupling to the order
parameter. Herein lies a crucial point: because strain, which represents the
internal degree of freedom of the lattice to distort, is a rank two tensor
quantity (strain is defined as the symmetric spatial derivative of a
displacement vector), it couples bilinearly to the quadrupoles, also rank two,
of the same symmetry. This coupling guarantees that for a finite applied
strain (induced by external stresses) of the appropriate symmetry there will
be a non-zero induced quadrupolar moment, and vice versa. Thus techniques in
which the material experiences strain are directly applying the conjugate
field for ferroquadrupolar phase transitions, and so are of considerable
interest for materials like TmAg2. Specifically, we use the AC elastocaloric
effect technique, in which AC strains are induced in a sample, and the
resulting temperature changes (in quasi-adiabatic conditions) arising from
electronic entropy shifts are measured, which recently was developed and
applied in studying pnictide superconductors, and apply it to TmAg2 in this
paper [17, 18].

The elastocaloric technique, described in detail in [17], probes strain
derivatives of the entropy, analogous to the magnetocaloric effect which
measures derivatives of the entropy with respect to magnetic field. Thus even
in the disordered state of a material, at temperatures above the relevant
phase transition temperature, if a finite order parameter is induced by a
strain of the relevant symmetry, then there will be a non-zero change in the
entropy and hence a potentially measurable change in the temperature of the
sample. In fact for phase transitions in which the terms (to second order) in
the Ginzburg-Landau free energy can be written as:

$F=a(T)\psi_{\Gamma}^{2}+\lambda\psi_{\Gamma}\varepsilon_{\Gamma}+\frac{1}{2}C_{\Gamma}\varepsilon_{\Gamma}^{2}+...$
(1)

where $\psi_{\Gamma}$ is the order parameter, $\varepsilon_{\Gamma}$ is the
corresponding strain of the same irreducible representation (irrep) as the
order parameter ($\Gamma$), $\lambda$ is the strain-order parameter coupling
constant, and $C_{\Gamma}$ is the appropriate elastic constant for the irrep
of strain, the elastocaloric effect, which is the derivative of the
temperature change of the sample with respect to an induced strain, can be
shown to probe the order parameter-strain susceptibility of the material:

$\frac{\partial
T}{\partial\varepsilon_{\Gamma}}=-\frac{T}{C_{p}}\frac{\partial
S}{\partial\varepsilon_{\Gamma}}=\frac{T\lambda^{2}\varepsilon^{o}_{\Gamma}}{C_{p}}\frac{d\chi_{\Gamma}}{dT}$
(2)

where Cp is the heat capacity of the sample,
$\chi_{\Gamma}=\frac{d\psi_{\Gamma}}{d\varepsilon_{\Gamma}}$, and
$\varepsilon^{o}_{\Gamma}$ is the offset strain of the sample, discussed
shortly. Note that this equation features partial derivatives with respect to
strains, and so each strain tensor element induced will have an associated
elastocaloric effect. Since strain tensor elements cannot be induced in
isolation experimentally, (a uniaxial stress will induce mixtures of the
relevant different irreps of strain based on the corresponding Poisson ratios)
[19], an elastocaloric measurement will probe an admixture of strain
derivatives of the entropy. However Eq. (1) is only relevant for anti-
symmetric strains (symmetric strains will have dominant linear terms in the
free energy without bilinear couplings), and so measurements that display a
linear relationship with the applied offset strain $\varepsilon^{o}_{\Gamma}$
(thus changing sign through zero strain) can be reasonably assumed to arise
from nematic susceptibilies.

For materials like the Ba(Fe1-xCox)2As2 family, in which
$\varepsilon_{B_{2g}}$ strain plays a dominant role in the entropy landscape,
the elastocaloric effect has been shown to be a sensitive probe in determining
the thermodynamic nematic susceptibility above the structural phase transition
[18]. We show for the case of the ferroquadrupolar TmAg2, in which the B1g
quadrupole orders at 5K, that the elastocaloric technique can provide an
isolated measurement of the B1g nematic susceptibility, as our measurement
aligns well with previous elastic constant measurements. We also demonstrate
by applying magnetic fields with both heat capacity and elastocaloric
measurements, that the ferroquadrupolar phase and nematic susceptibility in
TmAg2 are well described by a semi-classical mean-field transverse-field Ising
phase diagram in which a magnetic field oriented along the c-axis $H_{z}$ acts
an effective transverse field.

###  I.1 Model Hamiltonian for $4f$ degrees of freedom

We first introduce a model $4f$ Hamiltonian for TmAg2 neglecting metallic
degrees of freedom (besides the inclusion of a interaction between $4f$ sites
mediated via conduction electrons treated at a mean-field level), for which
the CEF spectrum has a ground state non-Kramer’s doublet which maps precisely
to the spin 1/2 Ising model.

TmAg2 is a rare-earth intermetallic material with tetragonal crystal structure
at room temperature (space group I4/mmm), and belongs to the MoSi2 structure
type [8, 9]. The $4f$ electrons of the trivalent Tm+3 ion form local
electronic multiplets with a total angular momentum quantum number J=6. The
2J+1=13 states are split by the surrounding crystalline electric field (CEF),
and heat capacity and inelastic neutron scattering measurements provide
evidence that the ground state is a $\Gamma_{5}$ doublet composed of $a|\pm
5\rangle+b|\mp 3\rangle+c|\pm 1\rangle$ states(where $|i\rangle$ denotes
$m_{j}=i$.) [8]. The nearest excited state is a $\Gamma_{1}$ singlet (composed
of $|\pm 4\rangle$ and $|0\rangle$ states) approximately 14K higher in energy
[8]. Most notably, this is the only metallic $4f$ tetragonal material known to
undergo solely a continuous ferroquadrupolar phase transition, (coupled with
an tetragonal to orthorhombic structural phase transition.) It occurs at 5K
with an $O_{2}^{2}$ ($B_{1g}$ symmetry) quadrupolar order parameter. This
phase transition is driven by local electronic degrees of freedom to break the
rotational symmetry of the lattice, and hence is an example of an electronic
nematic phase transition. No other phase transitions have been observed in
this material above 1.5K.

The operators $O^{2}_{2}$, $J_{z}$ (and the $B_{2g}$ Stevens operator
$P_{xy}$) map onto the the standard Pauli matrices $\sigma_{i}$ when applied
to the restricted subspace of the ground state $\Gamma_{5}$ doublet [2]. They
obey the same canonical commutation relations, and so at low enough
temperatures relative to the CEF excited states, and ignoring any
complications from the metallic surrounding lattice, $H_{z}$ should
theoretically serve as an effective transverse field to the spontaneous
nematic B1g order, and hence can tune the $4f$ sublattice to quantum
criticality.

A model Hamiltonian, considering applied magnetic fields along the c-axis, can
then be written down to describe the physics of the $4f$ sites in this
material, where only a single site is considered through a mean-field
approximation:

$\begin{split}H_{4f}=&H_{CEF}-B_{2}^{2}\varepsilon_{B_{1g}}O^{2}_{2}-K_{B_{1g}}\langle
O^{2}_{2}\rangle O^{2}_{2}-g_{z}J_{z}H_{z}\\\ \end{split}$ (3)

The nonperturbative HCEF describes the crystal electric field from the
surrounding atoms, and as previously stated produces a ground-state
$\Gamma_{5}$ doublet with an excited singlet at $\Delta_{1}=14K$, with other
excited states 50K and higher [8]. The second term is the bilinear
magnetoelastic coupling of the B1g quadrupole operator $O^{2}_{2}$
($J_{x}^{2}-J_{y}^{2}$) to the corresponding lattice strain (induced or
spontaneous)
$\varepsilon_{B_{1g}}=\frac{1}{2}(\varepsilon_{xx}-\varepsilon_{yy})$, and the
following term is a mean field quadrupole-quadrupole interaction term
originating from the RKKY interaction mediated by the conduction electrons.
Note that this Hamiltonian will produce the bilinear terms like those
described in the free energy model in Eq. (1), and hence the quadrupole-strain
susceptibility $\frac{d\langle O^{2}_{2}\rangle}{d\varepsilon_{B_{1g}}}$
should be measurable with the AC elastocaloric technique if it induces
$B_{1g}$ strain.

To model the effects of strain on this material, the operators in this
Hamiltonian were applied in the full 13 dimensional Hilbert space of the
single site $4f$ electronic multiplet, and Eq. 3 was self-consistently
diagonalized to obtain the thermal expectation value of the quadrupolar
operator. Quantities like the quadrupolar order parameter and the heat
capacity are derived from the resulting free energy and shown respectively in
Figures 2 and 3 as functions of temperature.

Figure 1: Crystal structure and $4f$ CEF spectrum for the Tm3+ ion. The
crystal structure is shown on the left, belonging to the space group I4/mmm
with tetragonal MoSi2 structure type. The silver atoms that surround Tm3+ ions
produce a crystalline electric field (CEF) with a D4h point group symmetry,
which splits the 13 fold degeneracy of the $4f$ electronic multiplet (J=6),
with previous neutron scattering measurements indicating an Eg ground state
and depicted excited states [8].  Figure 2: Schematic of $4f$ ground state
splitting and developing a quadrupolar order parameter. The top panel depicts
a schematic of the three lowest energy levels of the $4f$ electronic
multiplet, in which the ground state doublet splits spontaneously at 5K as
each site develops a quadrupolar order parameter, the magnitude of which is
shown in the bottom panel. The lighter color symbols in the energy level
diagram correspond to the energies of a site experiencing strains in the B1g
symmetry channel (similar in magnitude to the strains that were input to best
match the experimental heat capacity shown in Figure 3), and the corresponding
smearing of the transition manifests in a finite order parameter above 5K, as
shown by the dotted line in the bottom panel.  Figure 3: $4f$ Heat capacity of
TmAg2 compared to simulations with ”built-in” strains. The top panel depicts
the measured heat capacity for both TmAg2 and YAg2. The bottom panel displays
the difference of heat capacities of the two materials C${}_{p}^{4f}$, denoted
by blue dots. The small black dots are the results of simulated heat capacity
just considering the CEF energy spectrum and a mean-field quadrupolar
interaction term. The black dashes are simulated heat capacity results which
also include three $B_{1g}$ strains as fit parameters to account for the
inherent broadening even in the nominally free-standing sample. The phase
transition temperatures in the simulations were determined by the mean-field
interaction constants $B^{2}/C+K$, which were chosen so the system had a
critical temperature $T_{Q}=5K$. This simulated heat capacity is the linear
combination:
$0.634C_{p}(\varepsilon_{1})+0.23C_{p}(\varepsilon_{2})+0.136C_{p}(\varepsilon_{3})$
where $\varepsilon_{1}=0.007\%$, $\varepsilon_{2}=0.1\%$, and
$\varepsilon_{3}=0.7\%$. These values were chosen to mimic the distribution of
strains likely present in the material even in the absence of induced strains.

##  II Results and Discussion

###  II.1 Heat Capacity in zero magnetic field

Figure 4: Variation of the temperature dependence of the heat capacity with
applied magnetic field. Data are shown for fields ranging from 0-5T, with
vertical offsets proportional to the applied magnetic field for clarity (no
offsets for $H=0T$). TQ is marked by a black arrows, showing its progressive
suppression with increased magnetic field Hz Figure 5: Interpolated color map
of the temperature derivative of the $4f$ heat capacity in the H-T plane. The
dotted line shows a semi-classical transverse field Ising fit of a phase
boundary with $T_{Q}=5K$ at zero field and $H_{c}=3.2\pm 0.1T$ at 0K. White
dots depict the peaks of $dC_{p}/dT$ which are difficult to see given the
color shading.

Heat capacity measurements were performed for several samples with different
levels of disorder evident in the broadening of the phase transition. The
least disordered is presented here in the top panel of Figure 3. The
ferroquadrupolar phase transition manifests in a mean-field step at 5K, with
is consistent with a continuous phase transition mediated through long-range
interactions (here the lattice) increasing the effective dimension above the
upper critical dimension for the Ising model [14]. Note this is in contrast to
the heat capacity of YAg2 shown in the same panel, which only displays a
similar phonon contribution as the Y ion has an empty $4f$ shell. Subtracting
the two measured heat capacities will isolate the $4f$ contribution which we
call C${}_{p}^{4f}$. This quanitity is displayed in the panel below, and it
can be noted that the mean-field step signature is smeared, and that there is
a sizable contribution to the heat capacity above 5K. To attempt to explain
this we included results from the previously described simulation (thick
dashed line) with “built-in” strains that could represent macroscopic strains
locked in the growth process as well as microscopic strains induced by
impurities/ dislocation. Even from the calculation without these strains
included (thin dotted line), the excited CEF states clearly provide a
contribution to the heat capacity at temperatures above 5K.

###  II.2 Heat capacity in magnetic fields

The $C_{p}$ data from 0T to 5T is shown in Figure 4, which shows a smooth
suppression of $T_{Q}$ with magnetic field, as seen by the arrows. These
traces (along with higher field traces) were interpolated in both field and
temperature, and the numerical derivative was taken to create the color plot
shown in Figure 5. The derivative marks the ferroquadrupolar phase transition
quite well as a function of magnetic field down to around 3K, where the
signature becomes too small in magnitude compared to the background to track
accurately. The phase transition boundary is consistent with a semi-classical
transverse field Ising phase boundary with $T_{Q}(0)=5K$ and $H_{c}=3.2\pm
0.1T$. As the magnetic field was increased to progressively higher fields, a
Schottky feature became evident. This feature is consistent with an energy gap
that is linearly proportional to the applied magnetic field, which is expected
for the energy spectrum of an isolated magnetic doublet.

###  II.3 Elastocaloric measurements in zero magnetic field

Figure 6: Elastocaloric effect at different offset strains Plotted is the
magnitude of the temperature oscillations induced from a small AC strain
($\approx 0.01\%$) the sample experienced, which manifests from the
elastocaloric effect. Uniaxial AC stress was applied along (1 0 0) while
sweeping temperature from 20K to 2K, for different DC stresses which induced
strains $\varepsilon^{\prime}_{xx}$ ranging from 0.15% (in red) to -0.12% (in
green).  Figure 7: Elastocaloric effect vs. DC strain The elastocaloric effect
is plotted against the induced DC strain for different temperatures. The slope
increases as the sample is cooled, indicating the nematic susceptibility is
increasing towards 5K. The data is linearly proportional to the applied DC
strain at all temperatures measured, indicating the DC and AC strains induced
were well within the perturbative regime of the material.

As shown via free energy arguments made previously, the elastocaloric effect
is a sensitive probe of fluctuations which couple to the induced strain that
the sample is experiencing due to externally applied stresses. In particular,
for a nematic/quadrupole-strain susceptibility of the material, the
elastocaloric effect in the disordered phase will measure a signal
$\big{(}\frac{dT}{d\varepsilon}\big{)}$ proportional to the strain multiplied
by the temperature derivative of a nematic susceptibility (of the symmetry
channel of the induced strain) as described in Eq. 2.

In the case of the elastocaloric measurements on TmAg2 that appear in this
work, the relevant nematic/quadrupole-strain susceptibility that is measured
is the $B_{1g}$ susceptibility, $\chi_{B_{1g}}$. From the Hamiltonian in Eq.
3, a Curie-Weiss form can be expected for the $B_{1g}$ quadrupole-strain
susceptibility:

$\chi_{B_{1g}}=\frac{d\langle
O^{2}_{2}\rangle}{d\varepsilon_{B_{1g}}}=\frac{(B^{2}_{2})^{2}Q^{2}}{T-K_{B_{1g}}Q^{2}}$
(4)

where Q is the size of the saturated quadrupolar moment of the ground state.
The elastocaloric effect $\frac{dT}{d\varepsilon^{\prime}_{xx}}$ is displayed
in Figure 6, with symbol colors ranging from red to green representing tensile
to compressive induced DC strains. Within the range of the DC strains measured
the elastocaloric signal remains linearly proportional to strain at all
temperatures, signifiying the above equation accurately describes the data,and
that effects arising from $A_{1g}$ symmetry strains have minimal effect on
$\frac{dT}{d\varepsilon_{xx}}$. The response appears to diverge as temperature
approaches the phase transition at 5K, as expected if the nematic
susceptibility is Curie-Weiss like.

In Figure 7 the EC effect is plotted for different temperatures against the
induced DC strains. The slope of this data was extracted at fixed
temperatures, and to determine the nematic susceptibility that quantity was
normalized by the ratio $C_{p}(T)/T$ to obtain $\frac{d\chi}{dT}$ as shown in
Eq. (2). This quantity is plotted in Figure 8 against a quantity proportional
to the temperature derivative of $B_{1g}$ elastic constants data measured by
Ref. [8]. The proportionality constant is composed of two factors, one which
is an arbitrary constant A that takes into account the unknown strain
transmission rate of the elastocaloric measurements. The second is given by
$\frac{1}{2}(1+\gamma)^{2}$, which depends on the temperature dependent in-
plane Poisson ratio $\gamma$ of the sample, is the ratio of the approximated
$\varepsilon_{xx}^{2}$ to the actual $\varepsilon_{B_{1g}}^{2}$ induced. The
temperature derivative of the nematic susceptibility as obtained from the
elastocaloric effect vs. elastic constants softening nearly exactly align with
a proposed Poisson ratio which softens from a room temperature value of 0.3 to
the expected value of 1 at the phase transition. This provides further
evidence that the elastocaloric measurement is dominated by the B1g
quadrupolar susceptibility, as contamination from other symmetry modes would
provide a background or temperature dependence which deviates from the elastic
constant measurement. The constant A (proportional to the strain transmission
ratio squared) provides a best fit when around 0.2, which indicates a strain
transmission ratio of approximately 50%. This is well within the expected
strain transmission ratio of a hard relatively thick (60$\mu$m) sample at low
temperatures. Thus the nematic susceptibility as deduced from elastocaloric
measurements of this material is consistent with that obtained from elastic
constants. We note that both give an estimated Weiss temperature
$T^{*}=K_{B_{1g}}Q^{2}=2.7K$ (taking into account the softening of the lattice
in the elastocaloric effect, which artificially increases $T^{*}$), implying
the quadrupolar interaction terms (that drive the ferroquadrupolar phase
transition) which originate from the conduction electrons are nearly equal in
magnitude to those that are mediated by the lattice.

Figure 8: Comparison of ECE with elastic constants The blue points represent
the temperature derivative of the nematic susceptibility obtained from the
ECE. The red line is the temperature derivative of the B1g elastic constant,
measured previously in [8], multiplied by a proportionality constant dependent
on the in-plane Poisson ratio which accounts for the softening of the lattice
that manifests in a more sharply diverging ECE, and for the unknown strain
transmission ratio ($\varepsilon^{\prime}_{xx}$ vs. $\varepsilon_{xx}$) of
this experimental setup, assumed to be temperature independent.

###  II.4 Field dependence of elastocaloric measurements

Applying a magnetic field along the c-axis to the material in the disordered
state will induce a gap $\Delta_{H}$ between the two states of the ground
state doublet which will be proportional to the strength of the field. This
will result in a reduced nematic susceptibility, and if the Hamiltonian in Eq.
(3) is considered only in this restricted Hilbert space, $\chi_{B_{1g}}$ is
governed by the equation:

$\chi_{B_{1g}}=\frac{d\langle
O^{2}_{2}\rangle}{d\varepsilon_{B_{1g}}}=\frac{(B^{2}_{2})^{2}Q^{2}\tanh(\frac{\Delta_{H}}{k_{b}T})}{T-K_{B_{1g}}Q^{2}\tanh(\frac{\Delta_{H}}{k_{b}T})}$
(5)

This is the equation of the longitudinal order parameter susceptibility in the
presence of an applied transverse field which induces a gap in the paranematic
phase of magnitude $\Delta_{H}$.

Figure 9: Temperature dependence of the strain derivative of the elastocaloric
effect for different magnetic fields. The strain derivative of the
elastocaloric data (normalized by the fraction $\frac{T}{C_{p}}$ to obtain the
temperature derivative of the nematic susceptibility) is plotted for magnetic
fields applied along the c-axis from 0T (blue) to fields to near the critical
field of $3.32T\pm 0.1T$ (light blue), to fields above the critical field
(red). The black dashed lines represent best fits using the temperature
derivative of Eq.(6) with the multiplying constant fixed at the zero-field
value, and the varied parameters $\Delta_{H}(K)/2$ and $T^{*}(K)$ are plotted
in the offset panel vs. magnetic field.

The temperature derivative of the susceptibility as obtained from
elastocaloric measurements is plotted in Figure 9, with magnetic fields
ranging from 0T to 5T, with higher density of magnetic fields measured near
the critical field of $3.2\pm 0.1$T to investigate possible consequences of
quantum critical fluctuations. The results are described by Eq. (5), with the
fits at each field displayed as black dashed lines. These fits are
quantitatively accurate at fields even above the critical field, although we
note that in general the data is overdetermined by the free parameters in Eq.
(5), so we decided to keep $\lambda(B^{2}_{2}Q)^{2}$ fixed at the zero-field
value, and let $\Delta_{H}$ and $T^{*}=K_{B_{1g}}Q^{2}$ vary as a function of
field. As can be seen (inset to Figure 9), $T^{*}$ is approximately
independent of field, while $\Delta_{H}$ displays nearly linear behavior with
magnetic field, as expected from a simple polarized doublet ground state
model. On closer inspection, deviations from these semi-classical expectations
might be evident near the critical field (at which $T_{Q}$ is completely
suppressed). However, the present sparse data density and sample quality
preclude drawing any conclusions about whether these are signatures of quantum
criticality. Nonetheless at a qualitative level, the elastocaloric effect in
the paranematic phase of TmAg2 is well explained via the quadrupole-strain
susceptibility obtained from a transverse-field Ising Hamiltonian, with
magnetic field along the c-axis acting as a transverse field to the B1g
quadrupole order.

##  III Discussion

We have established through elastocaloric and heat capacity measurements that
the H-T phase diagram of TmAg2 is consistent with a semi-classical transverse-
field Ising model, with magnetic field along the c-axis acting as an effective
transverse field to the $B_{1g}$ quadrupolar order parameter.

This motivates the study of the thermodynamic susceptibilies and transport in
the proximity of the completely suppressed ferroquadrupolar phase transition.
While potential kinks/non-linearities in the fitting parameters of the
thermodynamic nematic susceptibility were seen in the proximity of the
critical field, the overdetermined nature of the fits makes this only a
suggestive signature at best. Rather, the more well-established experimental
trend is that the quadrupole-strain susceptibility followed the expected form
for a susceptibility in an spin 1/2 Ising model with magnetic field acting as
an transverse field over the entire field and temperature range studied.
Further lower-temperature measurements are clearly motivated, performed using
crystals with higher RRRs or sharper heat capacity features to resolve whether
any consequences of quantum criticality are evident in transport or
thermodynamic properties.

We nevertheless want to emphasize the novelty of both the application of the
elastocaloric effect on a $4f$ intermetallic material, as well as
investigating TmAg2 in the context of nematic quantum criticality. The AC
elastocaloric effect has recently been developed and applied to investigate an
iron-based superconductor [18], as well as explore antiferroquadrupolar order
in DyB2C2 [20]. We have shown with this work that it is also a sensitive probe
of low-temperature ferroquadrupolar phase transitions, and uniquely suited to
study materials with couplings to anti-symmetric strains. For TmAg2 we have
shown that a c-axis magnetic field is consistent with functioning as a tuning
parameter to obtain a quantum phase transition. As discussed in the
introduction, this implies that B2g strain, theoretically an analogous
transverse field along with magnetic field in the context of the $4f$ ground
state, could also be a tuning parameter to obtain a quantum phase transition.
A strain-tuned quantum phase transition in TmAg2 would be the first known case
of a QCP tuned by anti-symmetric strain, and motivates more strain-based
measurements and developing different low-temperature experimental setups
which allow larger strains.

##  IV Methods

TmAg2 single crystals were grown from a melt inside tantalum crucibles.
Stoichiometric amounts of thulium and silver were loaded into a crucible with
one end open, and these crucibles were then capped and sealed in a tetra-arc
furnace in an argon atmosphere, which was pumped and purged numerous times to
reduce contamination. These Ta crucibles were sealed under vacuum in quartz
tubes, then heated in a box furnace to 1190C, held for 12 hours, then cooled
for 100 hours to 800C (80K below the melting point of TmAg2 measured from a
differential scanning calorimeter), held there for another 100 hours to
increase crystal quality, and then cooled to room temperature. The resulting
melt had large sections with crystalline facets/features which could be broken
from the boule and oriented with single crystal X-ray diffaction.

Heat capacity measurements were performed in a Quantum Design PPMS using a
standard thermal relaxation technique. Apiezon N-grease was used to secure the
sample to the sapphire platform. A correlation was also noted between the
measured resistance ratio at 300K vs 2K (RRR) and the sharpness of the phase
transition signature in heat capacity, as well as the ratio of the magnitude
of the critical feature to the non-critical background. Only the measurements
of samples with the highest ratio of critical heat capacity to background and
RRR greater than 50 were measured for heat capacity and elastocaloric
measurements.

For the elastocaloric measurements, a commerical Razorbill CS-100 strain cell
was used to apply strain to the samples, which were polished and cut along the
(1 0 0) direction via wire-saw to be approximately 2mm x 0.04mm x 0.06mm in
size. The samples were secured between two sets of mounting plates using
Stycast 2850FT Epoxy, which were screwed into the strain cell, to have a gap
of approximately 1mm. An AC voltage of 20V RMS at 72 Hz was applied to the
outer piezoelectric (PZT) stacks of the strain cell, which corresponded to
applying an AC displacement of the sample of approximately 0.01% of its
length. This frequency was experimentally determined by measuring the
elastocaloric signal at 10K for frequencies in the range of 10-200 Hz, and
choosing the frequency with the largest response. This implied the frequency
was at the plateau of the relevant thermal transfer function, which did not
observably shift in the temperature range measured [17]. DC voltages were
applied to both the outer set of PZTs and the inner PZT to reach a strain
range of 0.3% at low temperatures. To approximate the strain the sample
experienced a capacitor built in the strain cell was measured to provide the
relative displacement of the sample plates, which was divided by the length of
the gap. This however only approximated $\varepsilon_{xx}$ of the sample as it
assumes a 100% strain transmission, and so we denote this quantity as
$\varepsilon^{\prime}_{xx}$ for the remainder of the paper.

The temperature fluctuations in the sample induced by the AC strain were
measured using a small commercial RuO2 thin-film resistor/thermometer (Vishay
# CRCW01002K43FREL) which was attached to the sample via a 50$\mu$m diameter
gold wire. The gold wire was silver pasted to the sample as well as the back
of the thermometer, to provide high thermal conductivity from the thermometer
to the sample. The ceramic substrate of the thermometer was polished (not
affecting the thin-film) to have dimensions of 150$\mu$m x 75$\mu$m x 20$\mu$m
to reduce its heat capacity. The temperature signal was measured at the sum of
this strain frequency and the thermometer excitation frequency, extracted with
an SRS860 lock-in using a similar AC demodulation technique as described in
Ref. [21]. To increase the signal to noise ratio the voltage from the
thermometer was measured using a custom Wheatstone bridge in which the 3 other
resistive components were identical resistors to the thermometer, and the
bridge was thermally anchored to the cell.

##  V Acknowledgements

We want to acknowledge fruitful conversations with E. Berg. This work was
supported by the Gordon and Betty Moore Foundation Emergent Phenomena in
Quantum Systems Initiative through Grant GBMF9068.

##  VI Author Contributions

E.R., M.I, conducted the measurements. E.R performed the calculations. E.R
grew the samples. I.R.F. oversaw the project. E.R., and I.R.F. wrote the
manuscript with input from all authors.

##  VII Competing Interests

The authors declare no competing interests.

##  VIII Data Availability

All data supporting the findings of this study are available upon request.

## References

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