[a]Joint Institute for Nuclear Research, 141980, Dubna, Russian Federation

<EMAIL_ADDRESS>
# Note on Intrinsic Resolution in Liquid Organic Scintillators

O. Smirnov

###### Abstract

Fluctuations in photon production in scintillator could contribute to the
total energy resolution of a scintillation detector. This contribution, called
intrinsic resolution (IR), is one of the factors limiting the total energy
resolution in detectors based on liquid organic scintillators (LSs). There are
very few experimental measurements of the IR in LSs available, and the
underlying physics is not completely understood. We propose a phenomenological
description of IR of LSs and systematize the data available using a single
universal parameter that characterizes IR. We show that all experimental data
within the model demonstrate the presence of extra smearing of energy
resolution due to IR with a typical value of $\simeq 2$% at 1-MeV energy
release. The model can be used to simulate the effect of IR in LS-based
detectors.

Liquid Scintillator, Intrinsic Resolution

##  1 Introduction

The statistical properties of a scintillation detector response were studied
by Breitenberger [1] and independently by Wright [2]. The relative variance
$v(Q)$ of a PMT signal with a mean number of photoelectrons (p.e.) $Q$,
registered in a scintillation event with an average number of produced photons
$\overline{N_{ph}}$ can be expressed as:

$v(Q)\equiv\left(\frac{\sigma_{Q}}{Q}\right)^{2}=\frac{1+v_{1}}{Q}+v(p)+(1+v(p))\left[v(N_{ph})-\frac{1}{\overline{N_{ph}}}\right].$
(1)

The parameter $v(N_{ph})$, of primary interest for this note, is the relative
variance of photon production. For normally distributed production of photons,
the term $v(N_{ph})=\frac{1}{\overline{N_{ph}}}$ and the last term vanish.

The parameter $v(p)$ characterizes the relative variance of photon detection
efficiency over all possible paths from the origin to detection, and the
parameter $v_{1}$ is the relative variance of a single photoelectron response
(SER) of a PMT. Sometimes, the excess noise factor (ENF) is used instead of
the $1+v_{1}$ term defined as $ENF=1+v_{1}$.

The formula (1) was derived for a single PMT. In the case of a detector with
many PMTs, a corresponding expression can be obtained with similar terms (see
e.g. [3, 4]).

The origin of the excess over normal variations (i.e., $\frac{1}{N_{ph}}$)
during photon production was widely discussed for non-organic scintillators
(see, e.g., [5]). This excess is usually attributed to scintillator non-
linearities in the energy response, which are, in turn, the result of Landau
fluctuations and secondary electron production ($\delta$-rays). The mechanism
is not confirmed by simulations for liquid organic scintillators [6]. The
authors refer to the fluctuation in the energy transfer process, which is much
less efficient compared to the one in non-organic crystals.

Note that, in contrast to all other parameters, the $v(N_{ph})$ parameter is
not universal as it is defined for a particular value of $\overline{N_{ph}}$
corresponding to the given energy of the particle. In this paper, we would
like to derive a universal parameter that will allow comparing experimental
results at various energies and photoelectron yields.

##  2 Phenomenological Model

In general, the parameter $v(N_{ph})$ should be measured for the energy of
interest. Nevertheless, some simple considerations can be used to understand
the energy dependence of this parameter. First, let us note that the total
amount of photons produced in an event with the energy release E could be
expressed as:

$N_{ph}=LY_{ph}\cdot E\cdot f_{NL}(E,...),$ (2)

where $LY$ is the relative photon (light) yield per MeV and $f_{NL}(E,...)$ is
the function describing non-linearity of the energy response, the notation "…"
is for a set of possible parameters, which are of no importance here and will
be skipped in notations. The light yield and non-linearity factor depend on
the type of interacting particle. In our study, we consider only electron
interactions. With low light intensities at a single PMT of the detector and
linearity of electronics, one can expect a linear dependence of the average
amount of the detected photoelectrons $Q$ on the number of average photons
produced in the interaction: $Q=c_{Q}\cdot N_{ph}$. From here on, we will
discuss the average values of the involved variables, if otherwise is not
explicitly stated. The total amount of the photoelectrons collected is:

$Q=LY\cdot E\cdot f_{NL}(E),$ (3)

where $LY\equiv LY_{ph}\cdot c_{Q}$ is the relative photoelectron yield per
MeV.

Let us introduce the relative variance of the number of produced photons
$v(N_{ph}(1))$ at a 1-MeV energy release for electrons 111Energy release
fluctuations for $\gamma$’s are different because of multiple Compton
scattering and scintillation non-linearities, that is why they are not
discussed here. We also do not discuss scintillations caused by
$\alpha$-particles.. The corresponding charge deposit can be expressed as
$Q_{1}=LY\cdot f_{NL}(1)$. The $Q_{1}$ value coincides with the relative
photoelectron yield of the detector if the non-linearity factor is normalized
in a way to provide unity at 1 MeV ($f_{NL}(E)\equiv 1$). We will keep the
notation $Q_{1}$ in order to underline the importance of the physical amount
of photoelectrons in the study; the photoelectron yield could be generally
defined in a way that complicates its usage for the calculations.

If the variance of the number of photons produced for a 1-MeV particle is
$\sigma_{N_{ph}}^{2}(1)$, then:

$\sigma_{N_{ph}}^{2}(E)\simeq\frac{N_{ph}(E)}{N_{ph}(1)}\sigma_{N_{ph}}^{2}(1),$
(4)

where the approximate equality sign is used to underline a possible non-
linearity of intrinsic smearing with respect to energy. If the hypothesis of
fluctuation in the energy transfer process is valid, the equality should be
quite precise. The relative variance of the number of the photons produced is:

$\split
v(N_{ph}(E))=\frac{\sigma_{N_{ph}}^{2}(E)}{N_{ph}(E)^{2}}=\frac{\sigma_{N_{ph}}^{2}(1)}{N_{ph}(E)\cdot
N_{ph}(1)}=\frac{N_{ph}(1)}{N_{ph}(E)}\frac{\sigma_{N_{ph}}^{2}(1)}{N_{ph}^{2}(1)}\equiv\\\
\equiv\frac{N_{ph}(1)}{N_{ph}(E)}\frac{N_{ph}(1)+\delta_{N_{ph}}^{2}(1)}{N_{ph}^{2}(1)}=\frac{1}{N_{ph}(E)}+\frac{N_{ph}(1)}{N_{ph}(E)}v_{1}^{int}.$
(5)

Here, we introduced $\delta_{N_{ph}}^{2}(1)$, an "extra" variance with respect
to the pure Poisson variance at 1-MeV energy release $N_{ph}(1)$, and defined
the relative extra variance at 1 MeV as
$v_{1}^{int}\equiv\frac{\delta_{N_{ph}}^{2}(1)}{N_{ph}^{2}(1)}$.

It is convenient to rewrite (5) through the charge $Q$ in order to use it in
(1):

$\split v(N_{ph}(E))-\frac{1}{N_{ph}(E)}=\frac{f_{NL}(1)\cdot LY_{ph}\cdot
1}{f_{NL}(E)\cdot LY_{ph}\cdot E}v_{1}^{int}=\\\ =\frac{LY\cdot
f_{NL}(1)}{Q}v_{1}^{int}\equiv\frac{Q_{1}}{Q}v_{1}^{int},$ (6)

where $Q_{1}$ is the charge corresponding to a 1-MeV energy release for
electrons. Thus, the term $\left[v(N_{ph})-\frac{1}{\overline{N_{ph}}}\right]$
in formula (1) can be substituted by:

$v_{int}\equiv\frac{Q_{1}}{Q}v_{1}^{int}.$ (7)

We get rid of $N_{ph}$ and its variance in the expression, introducing the
only universal parameter $v_{1}^{int}$. The value of the parameter $Q_{1}$
depends on LS properties and detector characteristics. The formula is
applicable to any type of particle, but the values of $v(p)$ and $v_{1}^{int}$
can differ for electrons and gammas for the same charge collected. Note that
$v(p)$ can be neglected due to the typically small value compared to unity;
e.g., for a 10% non-uniformity, we have $v(p)=0.01\ll 1$.

As one can see from (7), in fact, the extra variance due to the intrinsic
resolution contributes to the "statistical" term $\frac{1+v_{1}}{Q}$ as it has
the same energy dependence $\frac{1}{Q}$. Basically, one can use the
"efficient" $v_{1}$ (or ENF) parameter in the form
$v_{1}^{eff}=v_{1}+Q_{1}v_{1}^{int}$
($\text{ENF}_{eff}=1+v_{1}+Q_{1}v_{1}^{int}$).

So, the intrinsic resolution effect mimics statistical smearing. It is more
pronounced for detectors with larger photoelectron yields. Indeed, for a 1-MeV
energy release, the contribution of $v_{1}^{int}$ should be compared to
$\frac{1+v_{1}^{Det}}{Q_{1}}$ — the bigger the $Q_{1}$ parameter, the bigger
the relative contribution of IR.

Taking into account the low $v_{1}^{int}$ values of a few units of $10^{-4}$
(see the following discussion and final table 1), it is clear that experiments
with a high photoelectron yield are more suitable for the $v_{1}^{int}$
measurement. In fact, for the expected intrinsic resolution of 2% at 1 MeV and
the photoelectron yield of about 100 p.e./MeV, the extra term is
$10^{2}0.02^{2}=0.04$, a rather low one compared to the $1+v_{1}$ values. In
this case, one should provide at least a $\simeq 1$% precision measurement of
the statistical term in order to separate the $Q_{1}v_{1}^{int}$ contribution.

##  3 Direct Measurements of Intrinsic Resolution in Liquid Organic
Scintillators

Recently, two measurements have been performed using Compton spectrometers and
LAB-based scintillators. In paper [7], these were the measurements with the
137Cs monoenergetic $\gamma$-source. The reported value characterizing
intrinsic energy resolution is $v_{int}(50\text{ keV})=0.02\pm 0.005$. Since
$v_{int}=\frac{Q_{1}}{Q}v_{1}^{int}=\frac{LY\cdot 1}{LY\cdot 0.05\cdot
f_{NL}(E,kB)}v_{1}^{int}\simeq\frac{20}{0.9}v_{1}^{int}$, we obtain
$v_{1}^{int}=(0.0011\pm 0.0003)$ corresponding to a $(3.3\pm 0.4)$% intrinsic
resolution at 1 MeV. The value $f_{NL}(k_{B},E_{e}=0.05\text{ MeV})\simeq 0.9$
is derived from figure 3 of [7].

In the paper [6], the value of $(1.83\pm 0.06)$% is reported for monoenergetic
0.976- MeV electrons. The formula used for the analysis contains the sum of
terms attributed to all possible contributions, and the term responsible for
IR has a simple form, $\delta_{int}^{2}$. If we compare the formula with (1),
we can see that the factor $(1+v(p))$ is missing (note that in [6] $v(p)$ is
called $\delta_{uniformity}^{2}$), but due to the small value of
$\delta_{uniformity}^{2}=(0.89\times 10^{-2})^{2}$, factor can be neglected.
We can derive $v_{1}^{int}=(3.3\pm 0.2)\times 10^{-4}$ corresponding to a
$1.82\pm 0.06$% intrinsic resolution at 1 MeV.

##  4 Intrinsic Resolution in the Borexino Analysis

In contrast with the direct measurements performed with LAB-based LSs
discussed in the previous section, the Borexino detector used LS on the base
of pseudocumene (PC). We are interested in the analysis at low energies, where
the term responsible for IR appears in analytical expressions for energy
resolution. An important point in the Borexino analyses is the energy
estimator $N$, namely the amount of triggered PMTs in an event222PMT
occupancy, defined as the fraction of detected events
$\frac{N_{detected}}{N_{triggers}}$ in a series of trials, is genetically the
same estimator if applied to a single event in a many-PMT detector. The number
of trials in this case coincides with the amount of working PMTs. [8, 9, 10,
11, 12]. The choice of this estimator instead of the above-mentioned simple
p.e. counting is justified mainly by the better energy resolution achievable
in the high-segmented detector in the sub-MeV region. We would like to note
another advantage of using this energy estimator: it doesn’t depend on the
linearity of the PMT electronics channel as well as on the calibration
precision of the individual PMTs of the detector (conversion of a rough ADC
measurement into the amount of true photoelectrons), which is, in turn, a non-
trivial problem mainly due to the large amount of PMTs in this kind of LS
detector.

The broadening of resolution with respect to the models with no intrinsic
resolution was noticeable, so an extra term was included in the Borexino
analytical model. In order to convert the Borexino parameter into the one from
our phenomenological model, let us consider the smearing of the $N$ energy
estimator occurring due to intrinsic resolution.

First, we note that the variance of the average number of the detected
photoelectrons $\mu$ with each PMT of the detector is:

$v(\mu(E))=\frac{\mu_{1}}{\mu}v_{1}^{int},$ (8)

where $\mu$ is the average amount of p.e. detected in an event with the energy
E and $\mu_{1}$ corresponds to a 1-MeV energy release.

We are interested in the extra term in the expression for the signal variance
due to the intrinsic smearing. Let us consider the simplest case of a point-
like source at an ideal spherical detector’s centre. The variance of the
number of triggered PMTs for the events with the fixed average $\mu$ value
follows the binomial distribution:

$\sigma_{N}^{2}=N_{PMT}p_{0}(1-p_{0}),$ (9)

where $N_{PMT}$ is the total number of PMTs of the detector and $p_{0}$ is the
probability of the absence of the signal at a single PMT. We are interested
only in the IR effect, so we will not take into consideration other effects,
such as the electronics threshold effect or variance of the p.e. detection
efficiency through the photocathode and angles of photon incidence. In this
case, $p_{0}$ corresponds to the Poisson probability P(0), and we will use
$p_{0}=e^{-\mu}$ in calculations.

We will need the average values of $p_{0}$ and $p_{0}^{2}$ over all events and
the average values of $\mu$ distributed in accordance with the $\rho(\mu)$
distribution for further calculations. Assuming the normal distribution of
p.d.f. of the extra smearing $\rho(\mu)$ with the average $\mu_{0}$ and the
relative variance from (8), we obtain for $\overline{p_{0}}$:

$\split\overline{p_{0}}=\frac{1}{N_{events}}\sum_{i}^{N_{events}}e^{-\mu_{i}}\simeq\int
e^{-\mu}\rho(\mu)d\mu=e^{-\mu_{0}(1-\frac{\mu_{1}}{2}v_{1}^{int})},$ (10)

and for $\overline{p_{0}^{2}}$:

$\split\overline{p_{0}^{2}}=\frac{1}{N_{events}}\sum_{i}^{N_{events}}e^{-2\mu_{i}}\simeq\int
e^{-2\mu}\rho(\mu)d\mu=e^{-2\mu_{0}(1-{\mu_{1}}v_{1}^{int})}=\\\
=e^{-2\mu_{0}+2\mu_{0}\mu_{1}v_{1}^{int}}=e^{-2\mu_{0}+2\mu_{0}\frac{\mu_{1}}{2}v_{1}^{int}+2\mu_{0}\frac{\mu_{1}}{2}v_{1}^{int}}=\overline{p_{0}}^{2}e^{\mu_{0}\mu_{1}v_{1}^{int}}.$
(11)

We used (10) to obtain (11). Also, we replaced summation with integration,
assuming a large sample of events. Integration is performed over all possible
$\mu$ values, and the integral can be calculated analytically in the case of
the normal distribution $\rho(\mu)$.

In order to calculate the variance over a set of events, we write:

$\split\sigma_{N}^{2}=\overline{N^{2}}-\overline{N}^{2}=\overline{N_{PMT}^{2}(1-p_{0})^{2}+N_{PMT}p_{0}(1-p_{0})}-\overline{N_{PMT}(1-p_{0})}^{2}=\\\
N_{PMT}^{2}(1-2\overline{p_{0}}+\overline{p_{0}^{2}})+N_{PMT}(\overline{p_{0}}-\overline{p_{0}^{2}}-N_{PMT}^{2}(1-2\overline{p_{0}}+\overline{p_{0}}^{2})=\\\
=N_{PMT}^{2}(\overline{p_{0}^{2}}-\overline{p_{0}}^{2})+N_{PMT}\overline{p_{0}}(1-\overline{p_{0}}e^{\mu_{0}\mu_{1}v_{1}^{int}})=\\\
=N_{PMT}\overline{p_{0}}(1-\overline{p_{0}}e^{\mu_{0}\mu_{1}v_{1}^{int}})+N_{PMT}^{2}\overline{p_{0}}^{2}(e^{\mu_{0}\mu_{1}v_{1}^{int}}-1).$
(12)

The statistical part (the first term in 12) contains an extra factor
$e^{\mu_{0}\mu_{1}v_{1}^{int}}$ that can be neglected due to the small
$v_{1}^{int}$ value ($\mu_{1}=0.28$ and $\mu$ is well below $\mu_{1}$ for the
energy range of interest in the Borexino analysis). In this way, we can
separate an extra term responsible for intrinsic resolution:

$\split\sigma_{int}^{2}=N_{PMT}^{2}\overline{p_{0}}^{2}(e^{\mu_{0}\mu_{1}v_{1}^{int}}-1).$
(13)

Equation (12) from the Borexino paper [12] contains, in its turn, the
following term responsible for intrinsic resolution333The multiplier $10^{-2}$
is missing in the article cited, we put it back here; also, we skip the
irrelevant factor $f_{eq}$ as it appears because of the varying number of
PMTs, and it keeps the factor $v(N)$ stable with respect to the initial amount
of PMTs.:

$\sigma_{int}^{2}=10^{-2}v(N)\cdot N.$ (14)

The term responsible for intrinsic resolution in (12) is reduced to the one
used in the Borexino approximation for $\mu\ll 1$. The approximation is
reasonable, especially when taking into account that most of the statistics in
the Borexino low-energy analysis are concentrated at the end-point of the 14C
$\beta$-decay, i.e., 156 keV, and the corresponding $\mu$ value at a single
PMT of the total of 2000 is $\mu\simeq 0.156\frac{551}{2000}=0.04$ p.e.:

$\split N_{PMT}^{2}p_{0}^{2}(e^{\mu\mu_{1}v_{1}^{int}}-1)\simeq
N_{PMT}^{2}p_{0}\mu\mu_{1}v_{1}^{int}=N_{PMT}N\frac{p_{0}}{p_{1}}\mu_{1}\mu
v_{1}^{int}\simeq\\\ \simeq N_{PMT}N\frac{1-\mu}{\mu}\mu_{1}\mu
v_{1}^{int}=N(N_{PMT}\mu_{1})v_{1}^{int}=NQ_{1}v_{1}^{int},$ (15)

the parameter $v_{1}^{int}$ is related to the Borexino’s $v(N)$ as444Note that
$v(N)$ is used as a constant in the Borexino analysis.:

$v_{1}^{int}\simeq 10^{-2}\frac{v(N)}{Q_{1}}.$ (16)

The value of the parameter $v(N)$ reported in [12] is $11.5\pm 1.0$, and thus
we can derive $v_{1}^{int}=(2.2\pm 0.2)\times 10^{-4}$ using the value of
$LY=(551\pm 1)$ p.e./[MeVx2000 PMT] from [12] as $Q_{1}$. This value of
$v_{1}^{int}$ corresponds to the intrinsic resolution of $(1.5\pm 0.1)$% at 1
MeV.

In the earlier Borexino collaboration papers [9, 11, 10] devoted to the low-
energy analysis, the constant term in the form $\sigma_{int}^{2}$ was used
instead of the energy-dependent one. The values of the fit parameter are
reported in [13]: $\sigma_{int}=1.69\pm 0.23$ and $LY=420.1\pm 1.4$ p.e./MeV.
No systematic study of the parameters is available, so we will use these
values as the indicative ones. Similar to the previous case, the value of the
parameter corresponds to the effective energy close to the end-point of the
14C $\beta$-decay. The fit in the early Borexino low-energy studies was
performed with the energy estimator not corrected for the varying live PMT
amount, and the LY=420.1 p.e./MeV corresponds to the average number of PMTs
operating in this period. The parameter $\sigma_{int}$ is linked to our
universal parameter $v_{1}^{int}$ by the relation:

$\sigma_{int}^{2}=NQ_{1}v_{1}^{int}.$ (17)

The lower limit in the Borexino fit, N=60, corresponds to 165 keV [9]. In
order to estimate the effective N value, we should estimate the value of the
effective energy for the fraction of the 14C spectrum above the threshold.
Apparently, the value is below the end-point of 156 keV. Taking into account
that the end-point of the visible 14C spectrum is smeared up to 230 keV, we
can use as an estimate of the average value the scaled low bound of the
smeared spectrum: $\simeq 156\frac{165}{230}$ keV, or $\simeq 110$ keV. So,
for $E=0.11$ MeV $N=40$, we got $v_{1}^{int}\simeq(1.7\pm 0.5)\times 10^{-4}$.

##  5 Discussion of Results

All the data are compiled in table 1. As one can see, the data indicate the
presence of extra smearing due to intrinsic resolution at the level of
$\simeq(1\div 3)$%. The data from the experiments differ, but since
scintillator mixtures were not the same, it is difficult to draw further
conclusions: the difference in values could be due to LS cocktail components.
The two measurements of Borexino are in good agreement, though no systematic
uncertainty study is available. The value from the later Borexino paper should
be more precise as the parameterization used in the analysis follows the
energy dependence of the parameter, the earlier analysis implied an effective
value, and the conversion into the universal parameter needs precise
estimation of the effective N value at the tail of the 14C spectrum. The
measurement by Formozov [7] for LAB-based LSs has a higher value compared to
the one by Deng et al. [6], but this could be the result of different LS
cocktails as well. Also, we would like to point out the relatively close
values for the most precise measurements [6] and [12].

In conclusion, we would like to discuss the implications of our study for the
JUNO experiment [14] where energy resolution is crucial for the success of the
neutrino mass hierarchy measurement. Energy reconstruction in JUNO is provided
by machine learning algorithms [15]. Information from all PMTs is fed to the
algorithms, which should provide the optimal reconstruction at a given energy.
The achievable precision of reconstruction is limited by the detector
properties and can differ for the $N$ and $Q$ energy estimators, the machine
learning algorithm should choose the optimal one. For completeness, let us
consider both classes of energy estimators.

The inclusion of the IR contribution results in a modification of the
statistical part of the expression for the energy resolution for the energy
estimator based on the total charge collected:

$\sigma_{stat}^{2}(Q)=(1+v_{1}^{Det})Q+Q_{1}v_{int}^{1}Q=(1+v_{1}^{Det}+Q_{1}v_{int}^{1})Q,$
(18)

and the statistical term $R^{\prime}_{stat}$ of the energy resolution
accounting for intrinsic resolution can be expressed through the "no IR" term
$R_{stat}$ as
$R^{\prime}_{stat}=R_{stat}\cdot\sqrt{\frac{1+v_{1}^{Det}+Q_{1}v_{int}^{1}}{1+v_{1}^{Det}}}$.
Taking into account the expected LY=1345 p.e./MeV (the value is quoted for the
detector center [16]; for the whole detector, it should be admittedly higher,
but no data are still available, so we will use the value as a lower limit for
$Q_{1}$) and $v_{1}^{Det}=0.48$55515,000 MCP PMTs and 5,000 dynode PMTs are
ordered for JUNO [16]. The corresponding ENFs were reported in [17]: 1.55 and
1.17, respectively. Based on these values, we calculated $v_{1}^{Det}$ as the
weighted value., the estimated degradation of the statistical term is above
14% ($R^{\prime}_{stat}\geq 1.14R_{stat}$) for $v_{1}^{int}=3.3\times 10^{-4}$
.

In the case of using the energy estimator based on the number of triggered
PMTs, the extra term in resolution is quantified by expression 12, or its
approximation 15. The extra variance is:

$\frac{Q_{1}}{N}v_{1}^{int},$ (19)

using the low-energy approximation $N\simeq Q$, we got
$\frac{Q_{1}}{N}v_{1}^{int}\simeq v_{1}^{int}$ at 1 MeV, i.e., we have an
extra 1.8% that should be summed up quadratically with the statistical term of
the "standard" resolution at 1 MeV for the $N$ energy estimator.

The degradation in energy resolution could be crucial for the success of the
JUNO experiment, so further studies are needed. Based on our model, we can
indicate the following simple way of including intrinsic resolution into the
standard MC description of the detector: the total amount of photons generated
for the fixed energy release should be generated using a normal distribution
with an energy-dependent extra width corresponding to the intrinsic smearing:
$\sigma_{N_{ph}}^{2}=\frac{1}{N_{ph}}+\frac{N_{ph}(1)}{N_{ph}}v_{1}^{int}$.

Measurement | LS cocktail | rough value | $v_{1}^{int}$ | $R_{int}(1\text{ MeV})$
---|---|---|---|---
|  |  | $\times 10^{-4}$ | %
Formozov [7] | LAB+1.5 g/l PPO | $v_{int}(50\text{ keV})=$ | $11\pm 3$ | $3.3\pm 0.4$
|  | $0.02\pm 0.005$ |  |
Deng et al [6] | LAB+2.5 g/l PPO |  $1.83\pm 0.06$% | $3.3\pm 0.2$ | $1.82\pm 0.06$
| \+ 3 mg/l bis-MSB | @ 0.976 MeV |  |
Borexino [12] | PS + 1.5 g/l PPO | $v(N)=11.5\pm 1$ | $2.2\pm 0.2$ | $1.5\pm 0.1$
Borexino [13] | PS + 1.5 g/l PPO | $\sigma_{int}=1.69\pm 0.23$ | $1.7\pm 0.5$ | $1.3\pm 0.3$
Table 1: Available data on intrinsic resolution and their conversion to the
universal parameter $v_{1}^{int}$. The last column contains the relative
energy resolution at 1 MeV due to intrinsic smearing

##  6 Acknowledgements

This work was supported by the Russian Science Foundation, project no.
21-12-00063. The author is grateful to N. Mazarskaya for her valuable help in
preparing the manuscript.

## References

  * [1] E.Breitenberger, Scintillation spectrometer statistics, in Progr. in Nucl.Phys. vol.4 (1955) pp.56-94.
  * [2] G.T.Wright, Statistics of photomultiplier scintillation counters, J.Sci.Instr. vol.31 (1954) pp.377-381.
  * [3] O. Smirnov. ‘‘Energy and Spatial Resolution of a Large-Volume Liquid-Scintillator Detector,’’ Instruments and Experimental Techniques, Vol.46 No3 (2003) 327-344.
  * [4] G. Bellini, et al., Borexino collaboration, "Final results of Borexino Phase-I on low-energy solar neutrino spectroscopy", Phys. Rev. D 89, 112007, 2014.
  * [5] M. Moszyński et al.,"Energy resolution of scintillation detectors", Nuclear Instruments and Methods in Physics Research Section A, 805 (2016) 25-35.
  * [6] Y. Deng et al., "Exploring the intrinsic energy resolution of liquid scintillator to approximately 1 MeV electrons", JINST 17 (2022) P04018.
  * [7] A. Formozov on behalf of the JUNO collaboration, "Measurement of liquid scintillator nonlinear response and intrinsic energy resolution", 2019 J. Phys.: Conf. Ser. 1390 012111.
  * [8] G.Bellini, et al., Borexino collaboration, "Neutrinos from the primary proton-proton fusion process in the Sun", Nature, vol. 512 (2014) 383.
  * [9] O. Smirnov, et al., Borexino collaboration, "Measurement of Neutrino Flux from the Primary Proton–Proton Fusion Process in the Sun with Borexino Detector", Physics of Particles and Nuclei, 2016, Vol. 47, No. 6, pp. 995–1002.
  * [10] M. Agostini, et al., Borexino collaboration, "A test of electric charge conservation with Borexino", Phys. Rev. Lett. 115, 231802 (2015).
  * [11] M. Agostini et al., Borexino collaboration, "Limiting neutrino magnetic moments with Borexino Phase-II solar neutrino data", Phys.Rev.D 96, 091103(R) (2017).
  * [12] M. Agostini, et al., Borexino collaboration, "Simultaneous precision spectroscopy of pp, 7 Be, and pep solar neutrinos with Borexino Phase-II", PHYSICAL REVIEW D 100, 082004 (2019).
  * [13] Pablo Mosteiro, PhD Thesis. Princeton University, 2014.
  * [14] Fengpeng An, et al., JUNO Collaboration, "Neutrino Physics with JUNO", J. Phys. G 43 (2016) 030401.
  * [15] Zhen Qian, et al., "Vertex and energy reconstruction in JUNO with machine learning methods", NIM A, Vol. 1010, 2021, 165527.
  * [16] JUNO Collaboration, "JUNO physics and detector", Progress in Particle and Nuclear Physics, 123, 2022, 103927.
  * [17] A. Abusleme et al., JUNO Collaboration, "Mass testing and characterization of 20-inch PMTs for JUNO", Eur. Phys. J. C (2022) 82:1168.