# Measurements of low energy nuclear recoil quenching factors for Na and I
recoils in the NaI(Tl) scintillator

S. H. Lee  University of Science and Technology (UST), Daejeon 34113, South
Korea  Center for Underground Physics, Institute for Basic Science (IBS),
Daejeon 34126, South Korea  H. W. Joo  Department of Physics and Astronomy,
Seoul National University, Seoul 08826, South Korea  H. J. Kim  Department of
Physics, Kyungpook National University, Daegu 41566, Republic of Korea  K. W.
Kim<EMAIL_ADDRESS>Center for Underground Physics, Institute for Basic
Science (IBS), Daejeon 34126, South Korea  S. K. Kim  Department of Physics
and Astronomy, Seoul National University, Seoul 08826, South Korea  Y. D. Kim
Center for Underground Physics, Institute for Basic Science (IBS), Daejeon
34126, South Korea  University of Science and Technology (UST), Daejeon 34113,
South Korea  Y. J. Ko  Center for Underground Physics, Institute for Basic
Science (IBS), Daejeon 34126, South Korea  H. S. Lee<EMAIL_ADDRESS>Center for Underground Physics, Institute for Basic Science (IBS), Daejeon
34126, South Korea  University of Science and Technology (UST), Daejeon 34113,
South Korea  J. Y. Lee  Department of Physics, Kyungpook National University,
Daegu 41566, Republic of Korea  H. S. Park  Korea Research Institute of
Standards and Science, Daejeon 34113,South Korea  Y. S. Yoon  Korea Research
Institute of Standards and Science, Daejeon 34113,South Korea

###### Abstract

Elastic scattering off nuclei in target detectors, involving interactions with
dark matter and coherent elastic neutrino nuclear recoil (CE$\nu$NS), results
in the deposition of low energy within the nuclei, dissipating rapidly through
a combination of heat and ionization. The primary energy loss mechanism for
nuclear recoil is heat, leading to consistently smaller measurable
scintillation signals compared to electron recoils of the same energy. The
nuclear recoil quenching factor (QF), representing the ratio of scintillation
light yield produced by nuclear recoil to that of electron recoil at the same
energy, is a critical parameter for understanding dark matter and neutrino
interactions with nuclei. The low energy QF of NaI(Tl) crystals, commonly
employed in dark matter searches and CE$\nu$NS measurements, is of substantial
importance. Previous low energy QF measurements were constrained by
contamination from photomultiplier tube (PMT)-induced noise, resulting in an
observed light yield of approximately 15 photoelectrons per keVee (kilo-
electron-volt electron-equivalent energy) and nuclear recoil energy above 5
keVnr (kilo-electron-volt nuclear recoil energy). Through enhanced crystal
encapsulation, an increased light yield of around 26 photoelectrons per keVee
is achieved. This improvement enables the measurement of the nuclear recoil QF
for sodium nuclei at an energy of 3.8 $\pm$ 0.6 keVnr with a QF of 11.2 $\pm$
1.7 %. Furthermore, a reevaluation of previously reported QF results is
conducted, incorporating enhancements in low energy events based on waveform
simulation. The outcomes are generally consistent with various recent QF
measurements for sodium and iodine.

##  I Introduction

Currently, Weekly Interacting Massive Particles (WIMPs), deemed as one of the
most promising dark matter candidates [1, 2], have been extensively explored
by numerous experiments, yet with no conclusive findings [3, 4]. This has
prompted a shift in focus towards the low-mass dark matter realm, driving
increased interest in low energy events [5, 6, 7, 8]. This growing interest is
relevant not only to dark matter studies but also to neutrino experiments,
especially those involving coherent elastic neutrino-nucleus scattering
(CE$\nu$NS) [9]. In these experiments, neutrinos interact with target
materials, depositing less than a few keV nuclear recoil energies (keVnr),
emphasizing the importance of understanding low energy nuclear recoils.

Scintillating detectors detect nuclear recoil by measuring electron recoil
energy, a process that involves distinct scintillation phenomena different
from those associated with electron recoil energy [10]. Consequently, the
quenching factor (QF), indicating the ratio of nuclear recoils to electron
recoils, plays a crucial role in searches for WIMPs and CE$\nu$NS. Variations
in the QF have substantial implications for the interpretation of physics
outcomes [11, 12].

The NaI(Tl) crystal is widely used in applications such as WIMP dark matter
searches [13, 14, 15, 16, 17] and neutrino experiments [12]. Positive signals
reported by the DAMA/LIBRA experiment [18, 19, 13] have attracted attention,
but interpretations of these signals remain uncertain [20, 11, 21]. Although
the COSINE-100 data has ruled out specific interpretations [22], model-
independent data analysis is inconclusive [23]. The ANAIS-112 experiment
presents inconsistent results with DAMA/LIBRA at about 3$\sigma$ level [14],
but a potential difference in QFs is suggested by DAMA/LIBRA [13].

DAMA/LIBRA reported QF measurements using neutrons from a 252Cf source [24].
The measured responses were compared with the simulated neutron energy
spectrum to obtain QF values of 0.3 for sodium and 0.9 for iodine, assuming
independence from the energy of the recoiling nuclei. Recently, various QF
measurements from refined methods of NaI(Tl) crystals with mono-energetic
neutron beams have been reported [25, 26, 27, 28, 29, 30]. In these
measurements, the detection of an elastically scattered neutron at a fixed
angle relative to the incoming neutron beam direction provides energy-
dependent QF values significantly lower than DAMA/LIBRA’s QF results,
approximately 0.1 at 10 keVnr for sodium. Although different methods of
nuclear recoil calibration may contribute to this discrepancy, the crystal
dependence of the QF values is not definitively excluded. Since DAMA/LIBRA
reported their annual modulation results with a 0.75 keVee energy threshold
[31] corresponding to 2.5 keVnr (assuming 0.3 sodium QF of DAMA/LIBRA [24]),
understanding low energy nuclear recoil QF is essential.

The calibration of low energy nuclear recoil in NaI(Tl) crystals is highly
necessary, yet no measurements below 5 keVnr currently exist, primarily due to
noise induced by photomultiplier tubes (PMTs) contaminating a few
photoelectron events corresponding to about 0.5 keVee energy. We have
developed a novel crystal encapsulation technique aimed at enhancing the light
collection efficiency of NaI(Tl) crystals by directly attaching PMTs to the
crystal, resulting in nearly 50 % increased light yield [12]. This technique
has been applied to the NEON experiment, measuring CE$\nu$NS at a reactor,
yielding a consistently high light yield above 22 photoelectrons per keVee
(NPE/keVee) [12]. For a small-sized crystal with dimensions of 1.8 cm $\times$
1.8 cm $\times$ 1.4 cm, a comparable crystal encapsulation method was adopted,
achieving a 26 NPE/keVee high light yield. This allowed us to measure low
energy events below 0.5 keVee.

This paper presents measurements of the nuclear recoil QFs of a NaI(Tl)
crystal utilizing a high light yield NaI(Tl) crystal detector. Simulating the
scintillation waveform of NaI(Tl) crystals [32, 33], the characteristics of
low energy scintillation events are effectively investigated while mitigating
PMT-induced noise events. The event selection efficiency and the trigger
efficiency are assessed through simulated signal samples, reaching a 0.2 keVee
analysis threshold. This threshold enables us to measure the low energy QF as
low as the nuclear recoil energy of 3.8 keVnr for sodium and 6.1 keVnr for
iodine.

It is noteworthy that our prior measurements, conducted with a NaI(Tl) crystal
having a light yield of approximately 15 NPE/keVee [28], demonstrated similar
energy-dependent tendencies. However, the sodium QF values at 10 keVnr were
nearly 30% smaller compared to measurements by other research groups [26, 30].
With an enhanced understanding of low energy scintillation events and a
precise calibration of incident neutron energy, we identified a bias toward
lower QF values in our earlier measurement. Consequently, a thorough
reexamination of the data analysis for the previously reported data was
performed. The new measurement, deploying a high light yield NaI(Tl) detector,
along with the reevaluated results of the previous measurements, yields QF
values consistent with those reported by other research groups for both sodium
[26, 30, 29] and iodine [25].

##  II  Time of flight (TOF) measurements for incident neutron energy
calibration

Mono-energetic neutrons are produced through the deuteron-deuteron nuclear
fusion reaction using a DD109 neutron generator (Adelphi Technology, Inc.)
[34]. The generator tube was successively shielded by borated polyethylene (40
cm thickness) and high-density polyethylene (40 cm thickness), which complies
with safety regulations. Neutrons were emitted through a 3.5-cm-diameter hole
in the shield at a 90∘ angle relative to the deuteron beam. The entire
experimental setup was aligned in the direction of the neutron beam.

The incident neutron energy was determined using a 3He detector positioned in
front of the collimated hole when the DD109 generator was initially installed,
resulting in a measured neutron beam energy of 2.43 $\pm$ 0.03 MeV at an 85
keV deuteron energy. However, for previous QF measurements, a 60 keV deuteron
energy was utilized to address higher trigger rates caused by higher neutron
and x-ray fluxes at higher deuteron energy. Given the reported near
independence of neutron energy from the deuteron energy at the 90∘ angle [35,
36], the assumption was made that the neutron beam energy remains at 2.43
$\pm$ 0.03 MeV even when the deuteron energy is reduced to 60 keV.
Nevertheless, due to the volume of the deuteron plate and the NaI(Tl) crystal,
there could be a certain angle distribution slightly different from 90∘. For a
systematic understanding of the incident neutron energy, we employed the
neutron energy calibration method based on time-of-flight (TOF) measurement.

Two liquid scintillator (LS) detectors were installed using EJ-301 from Eljen
Technology, previously used for neutron tagging in QF measurements [28].
Adjacent to the collimated hole of the neutron generator, a one-inch diameter
LS detector, named LS1, was situated. Additionally, a second LS detector (LS2)
with a three-inch diameter was placed at a 53.1 $\pm$ 0.3∘ relative to the
neutron beam direction and 320.0 $\pm$ 2.0 cm distance, as depicted in Fig. 1.

Figure 1: Experimental setup of the incident neutron energy measurements using
the TOF method. Neutrons are coming out from the deuteron-based generator
inside the polyethylene shielding. One-inch diameter LS detector (LS1) is
placed in front of the neutron beam direction and a three-inch diameter LS
detector (LS2) is placed far from the one-inch detector to measure the TOF of
the scattered neutrons.

If a neutron interacts with LS1 and subsequently with LS2, a TOF of the
scattered neutron can determine the scattered neutron energy $E_{n^{\prime}}$.

$E_{n^{\prime}}=\frac{M_{n}}{2}\left(\frac{d}{c\Delta t}\right),$ (1)

where $\Delta t$ is the TOF between two LS detectors, $d$ is a distance
between two LS detectors, and $c$ is the speed of light. The incident neutron
energy $E_{n}$ is the sum of the recoil energy $E_{rec}$ in the LS1 detector
and $E_{n^{\prime}}$,

$E_{n}=E_{rec}+E_{n^{\prime}},$ (2)

where $E_{rec}$ is determined as follow,

$E_{rec}=\frac{2(1+A-cos^{2}\theta-
cos\theta\sqrt{A^{2}-1+cos^{2}\theta})}{(1+A)^{2}}E_{n}.$ (3)

Here, A is the atomic mass number of the recoiled nuclei, and $\theta$ is the
angle between the neutron beam direction and the scattered neutron direction.

The TOF with a deuteron energy of 100 keV was first measured. To assess the
TOF of the neutron and gamma events, we differentiate between the induced
particles based on their pulse shapes, taking advantage of the longer decay
time characteristic of the neutron events [37]. The pulse shape discrimination
(PSD) parameter is defined as a ratio of tail charge to total charge [28, 37].
As illustrated in Fig. 2, coincident neutron events and $\gamma$ events are
effectively identified based on the PSD parameter and different TOFs. The
measured TOF values are found to be 240.8 $\pm$ 0.3 ns and 6.8 $\pm$ 0.1 ns
for neutron and $\gamma$ events, respectively. With a distance between the two
LS detectors of 320.0 $\pm$ 2.0 cm, the TOF for gamma events should be 10.7
$\pm$ 0.1 ns. Therefore, the measured TOF is corrected by adding 3.9 $\pm$ 0.1
ns. The corrected neutron TOF is then determined to be 244.7 $\pm$ 0.3 ns.
Using Eq. 2 with Hydrogen interaction in the LS detector, the incident neutron
energy is evaluated as 2.48 $\pm$ 0.07 MeV, where the uncertainty accounts for
the uncertainties in the distance and angle.

Figure 2:  The TOF ($\Delta t$) of the scattered neutron in two LS detectors
as a function of the PSD parameter, the tail-to-total charge ratio. Neutron
and gamma events are well separated with the PSD parameter as well as TOF.
One-dimensional distribution of TOF ($\Delta t$) for each particle is placed
in insets.

The incident neutron energy was measured based on the TOFs of two LS detectors
for two more deuteron energies, specifically at 60 keV and 80 keV. The
resulting measured neutron energies are 2.32 $\pm$ 0.06 MeV and 2.41 $\pm$
0.06 MeV for 60 keV and 80 keV, respectively. All results are presented in
Fig. 3. Notably, the incident neutron energies exhibit a clear dependence on
the deuteron energies.

Figure 3: The measured neutron energies from the TOF measurements as a
function of deuteron energy (black dots) are shown together with 3He detector
measurement (red cross) from Ref. [28]. The linear fit to the data is
represented by a blue dashed line.

##  III QF measurement with an improved crystal encapsulation

###  III.1 Experimental setup

To maximize light collection of a NaI(Tl) crystal, a small-sized crystal with
dimensions of 1.8 cm $\times$ 1.8 cm $\times$ 1.4 cm is directly attached to
two 3-inch R12669 PMTs, as illustrated in Fig. 4. The crystal is grown from
the WIMPScint-III powder, produced by Alpha Spectra Inc., and originates from
the same ingot as the COSINE-100 crystal 6 and 7 [38]. Instead of having a
quartz window to encapsulate the crystal with copper or aluminum case, only a
1 mm thin optical pad connects the crystal to the PMTs optically. This
technique has already been proved to enhance light collection [39, 12]
efficiency. The crystal-PMTs assembly is housed in an aluminum casing with a
thickness of 5 mm.

Figure 4: (a) Schematic design of NaI(Tl) detector encapsulation, where the
PMTs are directly attached to the crystal. (b) Experimental setup of the QF
measurement. Monoenergetic neutrons from the DD109 neutron generator are
directed towards the NaI(Tl) detector. Various sizes of nine LS detectors are
strategically positioned around the NaI(Tl) crystal for neutron tagging
purposes. The one-inch LS detector at 10.4∘ was specifically installed for the
low energy nuclear recoil measurement below 5 keVnr.

The encapsulated crystal is oriented towards the neutron beams in front of the
collimated hole, positioned 1.8 m away as depicted in Fig. 4. We positioned
the detector to expose its smallest face, measuring 1.8 cm $\times$ 1.4 cm, in
the neutron beam direction to minimize multiple scattering of the incident
neutrons. For this new measurement, a fixed deuteron energy of 100 keV was
used, maintaining stable operation conditions of the DD109 generator. This
corresponds to an incident neutron energy of 2.48 $\pm$ 0.07 MeV as shown in
Fig. 3. A 5 cm thick lead shield was added outside the polyethylene shield of
the DD109 generator to mitigate radiated x-ray or other generator-related
backgrounds. Additionally, for background reduction, a 10 cm lead layer and a
20 cm polyethylene layer were employed to shield the NaI(Tl) crystal in the
direction of the neutron beam. A 3 mm opening was incorporated for controlled
exposure.

Nine EJ-301 LS detectors were deployed surrounding the NaI(Tl) detector for
neutron tagging. These LS detectors come in three different sizes: four with a
1-inch diameter, three with a 2-inch diameter, and two with a 3-inch diameter,
arranged from the beam direction to outer angles. Specifically, to measure low
energy nuclear recoils below 5 keVnr for sodium, one of the 1-inch diameter
detectors is placed in the co-linear beam direction at 10.4∘. The arrangement
of the DD109 generator, the NaI(Tl) crystals, and the nine neutron tagging
detectors is illustrated in Fig. 4.

A trigger is initiated when a signal from the PMTs in the NaI(Tl) crystal,
corresponding to one or more photoelectrons, occurs in each PMT within a 200
ns time window. In addition, hits from at least one of the neutron tagging
detectors are required for triggering. The triggered signals are then recorded
as 8 $\mu$s waveforms with a 500 MHz sampling rate, utilizing the same data
acquisition system employed in the COSINE-100 experiment [40].

###  III.2 Analysis

####  III.2.1 NaI(Tl) Crystal Calibration and Light Yield

The electron-equivalent energy scale of the NaI(Tl) crystal is linearly
calibrated using a single $\gamma$ line of 59.54 keV, achieved through
exposure to a 241Am radioactive source. To estimate the light yield, we first
identify a cluster of photoelectrons [41], and then assess the light output
using the ratio between the charge of the 59.54 keV events and that of a
single photoelectron [38]. In this study, a high light yield of 26.0 $\pm$ 0.7
NPE/keVee was obtained, facilitated by the novel crystal encapsulation
technique [39]. This represents around a 70% increase compared to the average
values of 15 NPE/keVee obtained from COSINE-100 detectors with the previous
design [38], enabling exploration below 0.5 keVee energy.

We routinely conducted 241Am calibration measurements every day to monitor the
stability of light yield and correct gain variations. Throughout the
approximately one-month measurement period, the detector module consistently
maintained a stable and high light yield.

####  III.2.2 Nuclear Recoil Event Selection

The discrimination of nuclear recoil events from electron/$\gamma$ background
was achieved by utilizing nine neutron tagging detectors. Two parameters were
introduced to identify nuclear recoil events: the charge ratio of the 50 to
200 ns (tail) to the 0 to 200 ns (total), a robust discrimination parameter
[37] leveraging the longer decay time of nuclear recoil events, as depicted in
Fig.5. Additionally, neutron events scattered in the NaI(Tl) crystal and
tagged by the LS neutron tagging detectors showed a clear coincident timing in
TOF between the NaI(Tl) crystal and the neutron tagging detector, as shown in
Fig.5. The combination of these two parameters facilitates the straightforward
selection of isolated neutron candidate events.

Figure 5: The tail to total charge ratio of the LS neutron tagging detectors
plotted against the TOF ($\Delta$t) between the NaI(Tl) and the LS neutron
detectors. Nuclear recoil events, characterized by a longer decay time in the
LS detector, exhibit higher values of the tail to total charge ratio. The
inset illustrates the one-dimensional TOF for the neutron event, highlighting
the clear coincident timing between NaI(Tl) and neutron tagging detectors.

####  III.2.3  Waveform simulation

In understanding the behavior of signals in low energy regions, we implemented
a waveform simulation developed to characterize NaI(Tl) scintillation events
recorded with the COSINE-100 and NEON DAQ systems [40] as detailed in Ref.
[33]. The simulation generates scintillation signals of the NaI(Tl) crystal,
tuned with measured data to account for characteristics ranging from single
photoelectrons to the scintillation properties of the NaI(Tl) crystal,
including rising and decaying times, as well as triggering and digitization
aspects from the DAQ system.

The waveform simulation was fine-tuned using the selected nuclear recoil
events discussed earlier. Extensive validation of the simulation was
conducted, covering the shape of single photoelectrons to the reconstructed
parameters of NaI(Tl) scintillation events, as presented in Ref. [33]. An
example of the meantime parameter, shown in Fig.6, demonstrates excellent
agreement between scintillation events in the data and the simulation.
Subsequently, the data from waveform simulation was utilized to evaluate
signal efficiencies based on trigger and event selection criteria.

####  III.2.4 PMT-induced noise rejection

While coincident neutron scattering events are selected based on the
aforementioned parameters, this sample may still contain PMT-induced noise
events. These events arise from incident neutrons scattering in PMTs or
crystal encapsulation material, and the scattered neutrons are subsequently
tagged by the neutron tagging detectors. Neutron scattering in the PMTs or
encapsulation materials can generate $\beta$/$\gamma$ particles that directly
strike PMT glasses, producing Cherenkov light. In this scenario, PMT-induced
noise events in the NaI(Tl) crystals and isolated neutron events in the LS
neutron tagging detectors are recorded.

To discriminate PMT-induced noise from NaI(Tl) crystal’s scintillation events,
we introduced a meantime parameter developed for event selections in
COSINE-100 data [42]. The meantime parameter represents the logarithm of the
combined charge-weighted mean decay time of each PMT, distinguishing events
based on pulse shape, such as fast-decaying PMT-induced noise events from
Cherenkov radiation in the PMT glasses. Figure 6 illustrates the meantime
parameter as a function of energy for the neutron-tagged events. PMT-induced
noise events in this data are clearly distinguishable when compared with the
waveform simulation data. It is noteworthy that contamination of PMT-induced
noise is more critical at low energy regions. Using the selection criteria
(solid line in Fig. 6) for scintillation events, we can efficiently remove
PMT-induced noise events for energies above 0.2 keVee, corresponding to 5 NPE.

Figure 6:  Distributions of the meantime parameter for the measured data (blue
points), which had a selection for the neutron scattering events, and the
waveform simulation data (red points) are presented. The green-solid line
indicates criteria for event selection to remove the PMT-induced noise events.

####  III.2.5  QF extraction

Events having coincidences with the neutron detectors, passing nuclear recoil
selections, and clearing noise rejection cuts are applied to measure the QFs
of the NaI(Tl) crystal. The selected data for the fixed angle of the neutron
tagging detector are compared with simulated energy spectra for various QF
values. The GEANT4-based simulation includes mono-energetic neutron generation
from the deuteron plate of the DD109 generator, the flight of neutrons to the
NaI(Tl) crystal, scattering, and tagging of the scattered neutron with the
neutron tagging detectors, with the geometry of the neutron measurement setup
shown in Fig. 4. Efficiencies from the trigger and event selection are
evaluated using waveform simulation data, as illustrated in Fig.7, and are
applied to the simulated energy spectra. The evaluated efficiency is about 0.3
at 0.2 keVee ($\sim$5 NPE) and reaches unity above 15 NPE, corresponding to
about 0.6 keVee. The nuclear recoil energy from the collinear angle of 10.4∘
is evaluated as 3.8 $\pm$ 0.6 keVnr. Minimization of $\chi^{2}$ between the
measured energy spectrum and the simulated spectra from various QF values is
performed. The best fit value is obtained from the minimum $\chi^{2}$, and the
uncertainty is derived from the $\Delta\chi^{2}$=1 range (inset) as 11.2 $\pm$
0.7 % as shown in Fig. 7.

Figure 7: The nuclear recoil energy spectrum of the NaI(Tl) crystal tagged by
the neutron detector at 10.4∘ corresponding to 3.8 $\pm$ 0.6 keVnr for sodium
is shown with black-dot points. The best fit spectrum of QF (11.2 %) from the
simulated spectra is overlaid with the red-solid line. An inset shows the
$\chi^{2}$ distribution as a function of the QF value with $\pm$1 $\sigma$
range (orange band) representing the uncertainty. The total efficiency (blue-
dashed line) was evaluated from the waveform simulation data and was applied
to the simulated energy spectra.

We extract the QF values for both sodium and iodine from nine neutron tagging
detectors. Due to relatively low energy nuclear recoils and low QF values,
iodine QFs are evaluated for scattering angles above 33.0∘. Table 1 summarizes
the measured QFs of sodium and iodine for nine neutron tagging detectors at
different angles. Measured QFs strongly correlate with the nuclear recoil
energy, with smaller values for lower recoil energies. The iodine QF is
considerably smaller than for sodium recoils, similar to previous reports [25,
28].

Table 1: Summary of the QF measurement with the high light yield NaI(Tl) detector. Nuclear recoil energies ($E_{nr}$) at each scattering angle are taken from the GEANT4 simulation, accounting the dueteron plate structure, geometric structures of the NaI(Tl) crystals and the neutron tagging detectors. Uncertainties from neutron energies, efficiencies, and detector geometries are included. Scattering | Na | I
---|---|---
angle (∘) |  $E_{nr}$ (keVnr) | QF (%) |  $E_{nr}$ (keVnr) | QF (%)
10.4 | 3.8 $\pm$ 0.6 | 11.2 $\pm$ 1.7 | $-$ | $-$
12.7 | 5.6 $\pm$ 0.8 | 13.8 $\pm$ 1.6 | $-$ | $-$
24.6 | 19.8 $\pm$ 1.9 | 15.6 $\pm$ 1.1 | $-$ | $-$
33.0 | 33.6 $\pm$ 2.9 | 19.7 $\pm$ 1.2 | 6.1 $\pm$ 0.5 | 4.9 ${}^{+\text{0.6}}_{-\text{1.8}}$
39.3 | 48.5 $\pm$ 3.7 | 18.7 $\pm$ 0.8 | 8.9 $\pm$ 0.7 | 5.0 $\pm$ 0.5
47.0 | 67.2 $\pm$ 6.1 | 19.7 $\pm$ 1.2 | 12.3 $\pm$ 1.1 | 4.7 $\pm$ 0.6
50.7 | 76.9 $\pm$ 6.9 | 20.1 $\pm$ 1.2 | 14.1 $\pm$ 1.3 | 5.3 $\pm$ 0.5
58.7 | 100.7 $\pm$ 9.2 | 22.7 $\pm$ 1.2 | 18.2 $\pm$ 1.7 | 5.9 $\pm$ 0.5
74.6 | 151.2 $\pm$ 14.2 | 24.5 $\pm$ 1.5 | 27.5 $\pm$ 2.8 | 6.8 $\pm$ 0.5

##  IV Revisit the previous QF measurement

The decision to revisit the previous QF measurement, reported in Ref.[28], was
prompted by the identification of an incident neutron energy calibration
issue, as detailed in Sec.II. During this revisitation, we employ a
methodology similar to that used for the new measurement, incorporating
advanced event selection and efficiency evaluation based on waveform
simulation.

The incident neutron energy from the previous measurement has been re-
evaluated and adjusted to 2.34 $\pm$ 0.06 MeV, differing from the initially
reported value of 2.43 $\pm$ 0.03 MeV, as depicted in Fig. 3. This revision
leads to a lower neutron energy and consequently, a smaller nuclear recoil
energy in the NaI(Tl) crystals. For instance, at the lowest scattering angle
of the previous measurement (13.2∘), the originally reported nuclear recoil
energy of 5.8 $\pm$ 0.4 keVnr has been corrected to 5.5 $\pm$ 1.1 keVnr.

The PMT-induced noise rejection cut, as detailed in Section III.2.4, utilizing
waveform simulation data was implemented. In comparison to the previous
selection, this updated criterion demonstrates approximately 20 % higher
efficiency for events below 1 keVee with minimal noise contamination. The
analysis threshold is set to 5 NPE corresponding to 0.35 keVee. This
enhancement enables the analysis of lower-energy events from smaller
scattering angles, which were challenging to extract QFs in the previous
measurement.

Assessing the trigger and selection efficiencies is crucial for extracting QFs
for low energy nuclear recoils. The efficiency evaluation is reassessed using
waveform simulation, and the efficiency is validated with calibration data.
During this examination, it is observed that a distinctly lower trigger
efficiency, particularly below 2 keVee ($\sim$30 NPE), was employed in the
previous analysis. This led to a substantial bias toward lower QF values in
the low energy nuclear recoil region.

In the previous measurement, we evaluated the trigger efficiency using an
independent setup with NaI(Tl) and LaBr3 crystals irradiated by multiple
$\gamma$-rays from a 22Na source [28]. The radioactive decay of 22Na emits
three $\gamma$-rays, including one at 1274 keV and two at 511 keV. The 511 keV
$\gamma$ from the LaBr3 crystal was tagged, and low energy events in the
NaI(Tl) crystals were identified as coincident Compton scattering events from
the other $\gamma$-rays. The measured trigger efficiency was estimated by
comparing the total events to the triggered events. However, our assumption
that all events in the NaI(Tl) crystals coincident with the 511 keV $\gamma$
in the LaBr3 crystal were pure scintillation events overlooked the possibility
of direct hits on the PMT glasses by $\gamma$ rays, which could generate fast-
decaying Cherenkov light. Analysis of the NaI(Tl) and LaBr3 setup data
revealed a sharp increase in the event rate below 1 keVee, even though 511 keV
coincident $\gamma$ was required in the LaBr3 crystal. Because of the
asymmetric behavior of the PMT-induced noise events, the trigger probabilities
of these events were significantly lower, resulting in a tendency toward lower
trigger efficiency, as illustrated in Fig. 8.

The trigger efficiency was calculated based on the waveform simulation and
applied to the QF estimation. For validation of the waveform simulation data,
the trigger efficiency was reassessed using the same 22Na calibration data of
the NaI(Tl) and LaBr3 setup. PMT-induced noise events were eliminated from the
sample, and the increased random event rate from energetic $\gamma$ source
irradiation was accounted for by excluding the contribution from random
coincidence events. As depicted in Fig.8, the trigger efficiency derived from
the waveform simulation aligns well with the reevaluated efficiency. However,
the trigger efficiency applied in the previous analysis was markedly lower
than that of the revisited analysis, as well as the waveform simulation data.

Figure 8: (a) Trigger efficiencies using a 22Na source with NaI(Tl) and LaBr3
coincident measurement data, comparing the previous analysis (red-solid line)
and the revisited analysis (points), are presented alongside the waveform
simulation data (dashed line). The revisited trigger efficiency aligns well
with the results from the waveform simulation data. (b) Energy distributions
of the nuclear recoil events tagged by the 16.4∘ neutron tagging detector,
corresponding to the nuclear recoil energy of 8.7 $\pm$ 1.3 keVnr for sodium,
are shown. The selected events using the previous analysis (red-solid line)
and the revisited analysis (blue-solid line) are compared with energy spectra
after the efficiency correction for the previous analysis (blue-circle points)
and the revisited analysis (red-square points). Due to the skewed trigger
efficiency of the previous analysis, the efficiency-corrected spectrum shows
an excess of events at low energy, resulting in biased low QF values in the
previous analysis.

Figure 8 illustrates the energy spectra of nuclear recoil events tagged by the
neutron detector at 16.4∘, corresponding to 8.7 $\pm$ 1.3 keVnr, selected
using the previous criteria (blue-solid line) and the revisited criteria (red-
solid line). Additionally, the energy spectra corrected by the evaluated
efficiencies for both the previous analysis (blue-circle points) and the
revisited analysis (red-square points) are presented. Due to the biased
trigger efficiency in the previous analysis, the QF value was lower, measuring
at 9.6 $\pm$ 1.6%. However, after revisiting the analysis method, the
corrected QF is 13.0 $\pm$ 2.2%.

Table 2 summarizes revisited QF results in comparison to the previous
measurements. The application of advanced event selection has led to the
recovery of a few small scattering angles, corresponding to lower recoil
energies. Slight increases in QFs with decreased incident neutron energy from
new calibration are observed for high scattering angles. Revisiting the
trigger efficiency at small scattering angles significantly increases the
measured QF values. These updated results align with the findings from the new
measurement of the high light yield crystal, as well as with measurements by
other groups [26, 29, 25] as shown in Fig. 9.

Table 2: Comparison of revisited nuclear recoil energies ($E_{nr}$) and QFs with previously reported results. The revisited outcomes consider the neutron beam energy of 2.34 $\pm$ 0.06 MeV, the application of enhanced event selections, and accurate efficiency evaluation through waveform simulation. | Scattering |  $E_{nr}$ (keVnr) | QF (%)
---|---|---|---
|  angle (∘) | Prev. | New | Prev. | New
Na | 13.2 |  5.8 $\pm$ 1.0 |  5.5 $\pm$ 1.1 | $-$ |  11.3 $\pm$ 2.2
| 16.4 |  8.7 $\pm$ 1.3 |  8.3 $\pm$ 1.4 |  9.6 $\pm$ 1.6 |  13.0 $\pm$ 2.2
| 21.3 |  14.8 $\pm$ 1.6 |  14.3 $\pm$ 1.8 |  11.3 $\pm$ 1.2 |  12.8 $\pm$ 1.6
| 26.6 |  22.7 $\pm$ 2.0 |  21.8 $\pm$ 2.5 |  14.1 $\pm$ 1.3 |  14.6 $\pm$ 1.7
| 31.0 |  30.1 $\pm$ 2.2 |  29.0 $\pm$ 2.8 |  17.2 $\pm$ 1.3 |  17.7 $\pm$ 1.7
| 38.2 |  46.1 $\pm$ 2.8 |  44.4 $\pm$ 3.8 |  17.3 $\pm$ 1.1 |  17.7 $\pm$ 1.7
| 45.0 |  62.6 $\pm$ 3.2 |  60.2 $\pm$ 4.7 |  18.1 $\pm$ 0.9 |  18.5 $\pm$ 1.5
| 51.3 |  78.9 $\pm$ 3.6 |  76.0 $\pm$ 5.7 |  21.3 $\pm$ 1.0 |  22.1 $\pm$ 1.7
| 59.0 |  102.7 $\pm$ 4.1 |  98.9 $\pm$ 7.1 |  22.1 $\pm$ 0.9 |  22.7 $\pm$ 1.6
| 74.7 |  151.6 $\pm$ 5.0 |  145.9 $\pm$ 10.0 |  22.9 $\pm$ 0.7 |  23.7 $\pm$ 1.6
I | 45.0 |  11.3 $\pm$ 0.6 |  10.8 $\pm$ 0.9 | $-$ |  5.3 $\pm$ 0.5
| 51.3 |  14.6 $\pm$ 0.7 |  14.1 $\pm$ 1.0 | $-$ |  5.2 $\pm$ 0.4
| 59.0 |  18.9 $\pm$ 0.8 |  18.2 $\pm$ 1.3 |  4.3 $\pm$ 0.4 |  5.5 $\pm$ 0.4
| 74.7 |  28.7 $\pm$ 1.0 |  27.6 $\pm$ 1.9 |  4.7 $\pm$ 0.2 |  5.4 $\pm$ 0.4
| 126.9 |  62.2 $\pm$ 1.5 |  59.9 $\pm$ 3.8 |  5.6 $\pm$ 0.2 |  5.7 $\pm$ 0.4
| 159.4 |  74.9 $\pm$ 1.6 |  72.1 $\pm$ 4.5 |  5.9 $\pm$ 0.2 |  6.2 $\pm$ 0.4
|
---|---
(a) | (b)
Figure 9:  (a) Results of the QF measurements for sodium using the high light
yield NaI(Tl) crystal (red-filled triangle) and the revisited analysis of
previously reported data (red-open triangle) exhibit consistency with recent
measurements from other groups [26, 29]. Similarly, results for iodine from
the new measurement (red-filled squares) and the revisiting of previous data
(red-open squares) also align with measurements from other study [25]. In this
new measurement, sodium and iodine QFs were obtained for as low as 3.8 keVnr
and 6.1 keVnr, respectively. (b) The measured QFs (red-open circle for sodium
and red-open squares for iodine), calibrated with 59.54 keV $\gamma$, are
modeled using modified Lindhard models (red-dotted lines) with their
corresponding uncertainty bands. Taking into account the non-proportionality
(nPR) of the NaI(Tl) scintillation process, the measured points are corrected
(blue-filled triangles for sodium and squares for iodine). The fitted models
are also adjusted accordingly (blue-dashed lines with uncertainty bands).

##  V Results and discussions

QFs for sodium recoils, spanning a broad energy range from 3.8 to 151.6 keVnr,
and iodine recoils ranging from 6.1 to 74.9 keVnr, have been successfully
obtained, extending this investigation into lower energy regions. Leveraging
the high-light-yield crystal, the achievement includes a remarkable
measurement of 3.8 keVnr, representing the lowest nuclear recoil measurement
for sodium to date. Through a thorough revisitation of previously reported
data, involving updates in neutron energy calibration, event selections, as
well as selection and trigger efficiency evaluations, the revisited QFs are
presented alongside the new measurements, as depicted in Fig.9 (a). These
results are compared with measurements from other research groups [26, 43,
25], showing consistent values. The QF values for each recoil angle obtained
in this study are detailed in Tables 1 and 2.

The QFs presented in Fig.9(a) from various experiments have been linearly
calibrated using either a 59.54 keV $\gamma$ line or a 57.6 keV $\gamma$ line.
However, discrepancies in QF values arise from the utilization of different
energy calibration methods involving distinct $\gamma$ energy points, as
extensively discussed in Ref. [30]. This discrepancy is primarily attributed
to the non-proportionality of NaI(Tl) crystals [44, 45]. To address non-
proportionality and ensure an accurate conversion between the measured charge
(light output) and deposited energy, it becomes imperative to implement an
energy-dependent calibration for both $\gamma$s and electrons. Our
investigation into the non-proportionality phenomenon involved a thorough
examination utilizing X-rays and $\gamma$-rays sourced from external
radioactive materials, as well as internal or cosmogenic contaminants present
in the COSINE-100 crystals [46]. This comprehensive study yielded an energy-
dependent calibration curve, enabling the recalculation of QF results. Figure
9(b) shows the QF values obtained in this study, accompanied by the
recalculated values considering the non-proportionality of the NaI(Tl)
crystals for both $\gamma$ and X-rays. These data points are modeled using the
modified Lindhard model [47] as described in Ref. [11]. The inclusion of non-
proportionality in the QF assessment reveals a notable increase in QF values,
particularly for the low energy region, underscoring the importance of using
appropriate QF values depending on the electron-equivalent energy calibration.

##  VI Conclusion

Measurements of low energy QFs for NaI(Tl) crystals were conducted using a
high light yield crystal of 26.0 $\pm$ 0.7 NPE/keVee, achieved through a novel
crystal encapsulation technique. The precise evaluation of QFs for low energy
nuclear recoils was facilitated by an in-depth understanding of low energy
scintillation events from waveform simulations. These measurements extended to
sodium QFs as low as 3.8 keVnr, marking the first measurement for sodium below
5 keVnr. Furthermore, a revisitation of previously reported data, with an
enhanced comprehension of incident neutron energy and a refined understanding
of low energy scintillation events through waveform simulation, was reported.
The measurements show general consistency with recent reports from other
research groups. These low energy QF measurements for NaI(Tl) crystals hold
relevance for future data analyses in the pursuit of dark matter and Coherent
Elastic Neutrino-Nucleus Scattering (CE$\nu$NS) studies with NaI(Tl) crystals,
such as the COSINE-100 and NEON experiments.

###### Acknowledgements.

We thank the IBS Research Solution Center (RSC) for providing high-performance
computing resources. This work is supported by the Institute for Basic Science
(IBS) under project code IBS-R016-A1.

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