# Structural properties of amorphous Na3OCl electrolyte by first-principles
and machine learning molecular dynamics

Tan-Lien Pham  Current address: Department of Chemistry and Biochemistry
Florida State University Tallahassee, FL 32304, USA  Multiscale Materials
Modeling Laboratory, Department of Physics, University of Ulsan, Ulsan 44610,
Republic of Korea  Mohammed Guerboub  Université de Strasbourg, CNRS, Institut
de Physique et Chimie des Matériaux de Strasbourg, UMR 7504, F-67034
Strasbourg, France  ADYNMAT CNRS consortium, F-67034 Strasbourg, France  Steve
Dave Wansi Wendj  Université de Strasbourg, CNRS, Institut de Physique et
Chimie des Matériaux de Strasbourg, UMR 7504, F-67034 Strasbourg, France
ADYNMAT CNRS consortium, F-67034 Strasbourg, France  Assil Bouzid  Institut de
Recherche sur les Céramiques, UMR 7315 CNRS-Univesité de Limoges, Centre
Européen de la Céramique, 12 rue Atlantis 87068 Limoges Cedex, France
Christine Tugene  Université de Strasbourg, CNRS, Institut de Physique et
Chimie des Matériaux de Strasbourg, UMR 7504, F-67034 Strasbourg, France
Mauro Boero  Current address: CNRS, Laboratoire ICube, Université de
Strasbourg, UMR 7357, F-67037 Strasbourg, France  Université de Strasbourg,
CNRS, Institut de Physique et Chimie des Matériaux de Strasbourg, UMR 7504,
F-67034 Strasbourg, France  ADYNMAT CNRS consortium, F-67034 Strasbourg,
France  Carlo Massobrio  Current address: CNRS, Laboratoire ICube, Université
de Strasbourg, UMR 7357, F-67037 Strasbourg, France  Université de Strasbourg,
CNRS, Institut de Physique et Chimie des Matériaux de Strasbourg, UMR 7504,
F-67034 Strasbourg, France  ADYNMAT CNRS consortium, F-67034 Strasbourg,
France  Young-Han Shin  Multiscale Materials Modeling Laboratory, Department
of Physics, University of Ulsan, Ulsan 44610, Republic of Korea  Guido Ori
Author to whom correspondence should be addressed<EMAIL_ADDRESS>Université de Strasbourg, CNRS, Institut de Physique et Chimie des Matériaux
de Strasbourg, UMR 7504, F-67034 Strasbourg, France  ADYNMAT CNRS consortium,
F-67034 Strasbourg, France

###### Abstract

Solid-state electrolytes mark a significant leap forward in the field of
electrochemical energy storage, offering improved safety and efficiency
compared to conventional liquid electrolytes. Among these, antiperovskite
electrolytes, particularly those based on Li and Na, have emerged as promising
candidates due to their superior ionic conductivity and straightforward
synthesis processes. This study focuses on the amorphous phase of
antiperovskite Na3OCl, assessing its structural properties through a
combination of first-principles molecular dynamics (FPMD) and machine learning
interatomic potential (MLIP) simulations. Our comprehensive analysis spans
models ranging from 135 to 3645 atoms, allowing for a detailed examination of
X-ray and neutron structure factors, total and partial pair correlation
functions, coordination numbers, and structural unit distributions. We
demonstrate the minimal, albeit partially present, size effects on these
structural features and validate the accuracy of the MLIP model in reproducing
the intricate details of the amorphous Na3OCl structure described at the FPMD
level.

##  I Introduction

Solid-state electrolytes are a focal point in the advancement of
electrochemical energy storage technologies, offering the potential to address
the safety and performance challenges inherent in traditional liquid
electrolytes [1, 2]. Notably, antiperovskite electrolytes, rich in Li and Na
with low melting points, have been identified as a promising category due to
their superior ionic conductivity and expedited synthesis processes [3, 4, 5,
6, 7]. Antiperovskite electrolytes based on X3OA (X= Li+ or Na+; A= halides
(Cl-, Br-, I-) or other anions (BH${}_{4}^{-}$, NO${}_{2}^{-}$)) have been
synthesized in various forms, including purely crystalline, glassy, and hybrid
amorphous/crystalline phases [3, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17]. In
particular, glassy Li3OCl- and Na3OCl-based antiperovskites have shown
remarkable conductivities of approximately $\sim$0.1 mS/cm at room
temperature, with glass transition temperatures between 390 and 450 K [18, 19,
20]. Also, amorphous Li3OCl has been used as a matrix to embed Li-La-Zr-Ta-O
garnet-type oxide particles, resulting in high room-temperature conductivity
of 0.2 mS/cm and an extensive electrochemical stability window up to 10 V. The
amorphous phase acted as a binder and filler, ensuring the formation of an
integrated composite solid electrolyte with a continuous, widespread ionic
conductive network. This significantly reduces the interfacial resistance with
lithium metal anodes [21, 22, 23, 24].
However, despite such enormous potential for applications [3, 16, 25, 26, 27,
28], a precise understanding of the structural and ion conduction mechanisms
in these glassy antiperovskite electrolytes is not been achieved yet. Indeed,
recent literature on antiperovskites for solid-state batteries has highlighted
the limited scope of structural characterizations available, underscoring the
urgent need for a quantitative structural assessment to prevent
misinterpretation of the relationship between structure and performance [13].
While the structure and ion dynamics in crystalline compounds are well
understood, amorphous antiperovskites await a clearcut structural
characterization. In this context, resorting to atomistic modeling has proved
worthwhile to better understand the structural and transport properties of
solid-state electrolytes [29], being this approach complementary to
experimental efforts.
A revealing example has been provided by our recent study of H-free sodium
oxyhalide Na3OCl and hydroxyhalide antiperovskites Na3-xOHxCl ($x$ = 0.5, 1)
using first-principles molecular dynamics (FPMD). Intricate details of their
amorphous structure, the dynamics of Na ions, the role of H atoms, and their
collective impact on ionic A revealing example has been provided by our recent
study of H-free sodium oxyhalide Na3OCl and hydroxyhalide antiperovskites
using first-principles molecular dynamics (FPMD). Intricate details of their
amorphous structure, the dynamics of Na ions, the role of H atoms, and their
collective impact on ionic conductivity have been highlighted [30]. Due to
intrinsic limits of our study based on a relatively small computational cell
of 135 atoms, an extension to large sizes is appropriate to estimate size-
dependent effects and obtain a more robust assessment of structural and
dynamical properties, fully devoid of artifacts that might characterize the
behavior of minimal size models. With this purpose in mind we focus here on
amorphous Na3OCl electrolyte by harnessing atomic-scale modeling through FPMD
and classical MD simulations driven by a machine learning interatomic
potential (MLIP). We considered models containing up to 3645 atoms fully
representative of experimental behaviors.

To substantiate our structural analysis, we conduct an in-depth comparison
with experimenta X-ray and neutron structure by considering also the total
pair correlation functions. While X-ray probing plays a crucial role, neutron
studies are unavoidable to investigate quantitatively antiperovskites,
especially due to their ability to detect low atomic number (Z) elements,
which are beyond the capabilities of X-ray techniques using synchrotron
sources [31, 32, 33]. Furthermore, neutron diffraction and scattering studies
are gaining increased interest for oxyhalide materials, as they provide
insights into the arrangement of sodium and oxygen atoms, well beyond the
level of detail achievable through X-ray methods [31, 32].
The paper is structured as follows: Section II outlines the FPMD and MLIP
modeling techniques and the systems studied, including the construction of the
FPMD database and the evaluation of the developed MLIP against DFT data.
Section III is devoted to the FPMD and MLIP results analysis, providing the
structural properties such as structure factors and total and partial pair
correlation functions, coordination numbers, and structural units
distributions. These findings are thoroughly analyzed and compared with
existing experimental and modelling data for both Na3OCl- and Li3OCl-based
systems. Concluding remarks are presented in Section IV.

##  II Computational methodology and models

###  II.1 First-principles molecular dynamic simulations

The Car-Parrinello (CP) method [34] was used to produce trajectories of the
targeted Na3OCl system. The exchange-correlation functional selected is the
generalized gradient approximation (GGA) proposed by Perdew, Burke and
Ernzerhof (PBE) [35]. The valence-core interaction was described by numerical
norm-conserving Troullier-Martins (TM) [36] pseudopotentials for all elements
(Na, O, and Cl). In the case of Na, semi-core states were included to ensure a
good description of the energetics and electronic features dependent on its
cationic nature. This amounts to electronic configurations [He] 2s2, 2p6, 3s1
for Na; [He] 2s2, 2p4 for O; and [Ne] 3s2, 3p5 for Cl. Valence electrons are
treated explicitly and represented on a plane-wave basis set with the sampling
of the Brillouin restricted to the $\Gamma$ point. A fictitious electron mass
of 600 a.u. and a time step of 0.12 fs ensured optimal conservation of the
constants of motion. FPMD simulations were performed in the canonical NVT
ensemble with the ionic temperature controlled with a Nosé-Hoover [37, 38, 39,
40] thermostat chain. [41] For the simulations performed at high temperature
($>$1000 K) we used a Blöchl-Parrinello electronic thermostat [42] with a
target kinetic energy of 0.1 a.u. to control the fictitious motion of the
electronic degrees of freedom. In our previous work [30], the Na3OCl glass
model was generated by quenching from the melt, starting from an initial
configuration consisted of a Na3OCl crystal cubic unit cell replicated
$3\times 3\times 3$ times to obtain a 135 atoms model (81 Na, 27 O, and 27 Cl)
in a cubic simulation cell of side 13.6148 Å [43]. The initial box was
expanded by 15% with respect to crystal density in order to ease melting at a
high temperature. The model underwent a thermal cycle via canonical NVT
simulations, from T = 1200 K to T = 300 K along and finally equilibrated at T=
300 K for about 70 ps (see ref. [30] for details). At the end of the cycle of
T = 300 K, the density of the system was set to 2.04 g/cm3 in order to reduce
the residual stress to values very close to 0 GPa. Along the thermal cycle the
energy cutoff for the plane wave expansion was REc= 80 Ry. In this study, we
generated two novel amorphous models of Na3OCl, consisting of 135 and 405
atoms, respectively, each possessing a density of 2.04 g/cm3. Both models
underwent a FPMD controlled thermal cycle following the following protocol: 3
ps at T = 300 K, 3 ps at T = 600 K, 3 ps at T = 900 K, 25.3 ps at T = 1200 K,
22 ps at T = 900 K, 22 ps at T = 600 K, and 30 ps at T = 300 K. The phase at T
= 1200 K exhibited a diffusion coefficient of the order of 10-5 cm2/s,
indicative of a complete loss of memory of the initial configuration.
Subsequent sections present the atomic structure results, reported as time-
average values over the last 20 ps of the thermal cycles at T= 300 K.
Specifically, the structural properties for the 135-atom FPMD size model
(denoted as FPMD135 hereafter) represent the average of results from the new
135-atom model produced in this work and those from ref. [30]. Similarly, the
405-atom model results correspond to the model produced in the present work,
identified as the FPMD405 model. For all the simulations performed here, we
used the developer version of the CPMD package [44].

###  II.2 DFT-FPMD database composition

We build a database made of reference configurations on the desired regions of
phase space and associated quantum mechanical data (energies, forces and
virial stresses) by extracting 400 representative configurations from the FPMD
trajectories obtained at different temperatures (1200, 800, 450 and 300 K) for
the 135-atoms model produced in ref. [30]. This amounts to a total of 400
total energy values, 162000 force components and 3600 virial components. In
order to achieve a good accuracy of the database, we recomputed DFT energies,
forces, and virials for all the configurations at an energy cutoff of 160 Ry.
The built database is then split in 50:50 for training and testing sets.

###  II.3 Machine learning interatomic potential: model fitting and training
database

We employ a kernel-based MLIP method using the Gaussian Approximation
Potential (GAP) scheme, enabling the learning and reproduction of smooth,
high-dimensional potential energy surfaces through the interpolation of DFT
data [45, 46, 47, 48]. This MLIP scheme has demonstrated success in various
applications, encompassing liquids, crystalline phases, as well as mono and
binary glasses [49, 50]. However, its application to ternary amorphous systems
has been relatively limited so far. In the subsequent section, we briefly
outline key aspects of the GAP methodology. Additional details regarding the
GAP schemes can be found in dedicated reviews [45, 48, Der2021, 52]. The GAP
scheme used in the present work is based on the system’s total energy
expressed as the combination of a repulsive two-body potential and a many-body
kernel based on the Smooth Overlap of Atomic Positions (SOAP) descriptor [53].

$E\equiv\sum\limits_{i<j}^{n}\mathbb{V}^{(2)}(r_{ij})+\sum\limits_{i}^{n}{\epsilon(\mathbb{q}_{i})},$
(1)

where, $i$ and $j$ index the atoms in the atoms, $V^{(2)}$ is the 2-body
repulsive pair potential, $r_{ij}$ is the distance between atoms $i$ and $j$,
and $\epsilon$($\mathbb{q}_{i}$) is the energy contribution of the $i^{th}$
atom and $\mathbb{q}_{i}$ is a descriptor characterizing the local atomic
environment around the $i^{th}$ atom. The local atomic energy contribution
$\epsilon$($\mathbb{q}_{i}$) is given by a linear combination of kernel
functions,

${\epsilon(\mathbb{q}_{i})}=\sum\limits_{j}\alpha_{j}K(\mathbb{q}_{i},\mathbb{q}_{j}),$
(2)

where, $K$ is a fixed kernel function quantifying the degree of similarity
between atomic environment described by $q_{i}$ and $q_{j}$. The coefficients
$\alpha_{j}$ are determined by a regularised linear regression of the
energies, forces and virials of the system computed with Eq.(2) that are
parametrized by these $\alpha_{j}$ values, and compared to the corresponding
energies, forces and stresses computed by DFT for the same structures. In
SOAP, the local environment of the $i^{th}$ atom is represented by its
neighbour density function within a given cutoff $r_{cut}$,

$\rho_{i}(\mathbb{r})=\sum\limits_{j}f_{cut}(r_{ij})exp\left[-\frac{(\mathbb{r}_{i}-\mathbb{r_{ij}})^{2}}{2\rho^{2}_{at}}\right],$
(3)

where, $\mathbb{r}_{i}$ denotes the position vector of atom $i$ and
$\rho_{at}$ controls the smoothness of the potential. Here, $f_{cut}$ is a
cutoff function going smoothly to 0 at $r_{cut}$ [54]. Atomic neighbor
densities are then expanded in a local basis set of orthogonal radial basis
functions $g_{n}$(r) and spherical harmonics $Y_{lm}$ as:

$\rho_{i}(\mathbb{r})=\sum\limits_{n<n_{max}}\sum\limits_{l<l_{max}}\sum\limits_{m=-l}c^{i}_{nlm}g_{n}(r)Y_{lm}\left(\mathbb{\hat{r}}\right),$
(4)

where $c^{i}_{nlm}$ are the expansion coefficients. The spherical power
spectrum of these expansion coefficients then forms the descriptor as:

$(\mathbb{q}_{i})_{nn^{\prime}m}=\frac{1}{\sqrt{2l+1}}\sum\limits_{m}(c^{i}_{nlm})^{*}c^{i}_{n^{\prime}lm},$
(5)

which is translational, rotational, and permutational invariance. For the
many-body SOAP descriptor, the dot product kernel is determined by

${K(\mathbb{q}_{i},\mathbb{q}_{j})}=\left|-\frac{(\mathbb{q}_{i}\cdot\mathbb{q_{j}})}{|\mathbb{q_{i}}|\cdot|\mathbb{q_{j}}|}\right|,$
(6)

The positive integer $\zeta$ allows to improve the sensitivity of the kernel
and also increase the body order of the model. To reduce the computational
cost, a sparsification method is used for the SOAP kernel. Overall, the SOAP
descriptor represents atomic geometries by using a localized expansion of a
Gaussian-smeared atomic density. In so doing, the estimation of a specific
physical property is broken down into individual contributions centered on
atoms. These contributions effectively capture correlations between atoms
inside each localized environment. The assumption of locality, justified by
the nearsightedness principle [55] of electronic matter, is very advantageous
as it effectively reduces the complexity of the regression task, often
encountered in machine learning schemes. The SOAP descriptor facilitates a
thorough characterization of short- to medium-range order, allowing the
identification of features that assess varying degrees of (dis)order in a
given material [46, 30].
The hyperparameters used in this work for constructing the descriptors, the
kernel functions, and training a GAP are summarized in Table S1 of the
Supplementary Material.

Figure 1: Top: Scatter plots of DFT-computed and GAP-predicted total energies
(left), forces (centre) and virials (right) for the training and testing
(insets) sets of 200 configurations each. Bottom: Cumulative error
distributions: a given point ($x$,$y$) on the curve indicates that $y$ percent
of all structures have an error equal to or below $x$. Insets: Absolute errors
distributions of the respective quantities obtained for the training and
testing sets.

###  II.4 MLP-GAP model performance assessment: errors for testing versus
training sets

We here assess the validation of the accuracy of our GAP model based on the
DFT-FPMD reference data. The training and testing datasets contain the energy,
forces and virial components of 200 configurations each. As shown in Fig. 1,
the total energies, atomic forces and virial as predicted by our model are
compared with those from DFT-FPMD. Our GAP model successfully reproduces the
energies with a low mean average error (MAE) of 1.59 eV/atom for testing
datasets, significantly below the commonly referenced threshold of 5 meV/atom,
which is often cited as indicative of a high-performing MLIP [56, 57, 58].
Forces in the datasets are predicted with a MAE of 0.04 eV/Å whereas virials
are predicted with a MAE of 3.64 eV, respectively. The results demonstrate
that our GAP model is a good representative of the first-principles potential
energy surface for amorphous Na3OCl. Figure 1 displays errors through counting
and cumulative distribution plots. The counting distribution reveals a very
tight overlap, while the cumulative distribution shows curves closely aligned
for forces and slightly shift to the right and downward (indicating slight
increased confidence) when comparing predictions for energy and virials from
training and testing datasets. Overall, across all three observables, we note
an exceptionally close match, highlighting the smooth performance and first-
principles accuracy of the developed MLIP-GAP.

###  II.5 MLP-GAP molecular dynamics simulations

We exploited the fitted GAP potential to produce new models of amorphous
Na3OCl by following a thermal cycle via canonical NVT simulations, according
to the following protocol: heating up from T = 300 K to T = 1200 K along 5ps,
melting at T = 1200 K for 100ps, cooling from T= 1200 K to T = 300 K with a
cooling rate of 4 K/ps, and final equilibration at T = 300 K for 200ps.
Subsequent sections present the structural analysis, reported as time-average
values over the last 100 ps of the thermal cycles at T= 300 K. The equations
of motion were integrated by using a timestep of 0.5-1 fs and NH thermostat
was used as implemented in the LAMMPS code [lammps22]. With such procedure we
produced MLIP-GAP models with the same density (2.04 g/cm3) for following
sizes: 135 atoms model (hereafter denoted as GAP135 model), 405 (GAP405), 1080
(GAP1080) and 3645 (GAP3645) atoms models. GAP135 was averaged over 11 replica
run whereas the other GAP models were averaged over 2 run. Figure 2 shows
snapshots and details of the models simulated by FPMD and GAP. The GAP models
obtained have been employed to reinforce the structural findings of amorphous
Na3OCl derived from FPMD. This is achieved by leveraging large-scale models
refined to DFT accuracy, thereby highlighting any potential size-related
effects encountered. Wherever pertinent, the outcomes of the GAP models are
thoroughly examined and discussed within the paper.

Figure 2: (Color online) Supercell amorphous models of Na3OCl simulated by
FPMD (135 and 405 atoms) and GAP (135, 405, 1080 and 3645 atoms). Color
legend: Na, yellow; O, red; Cl, green. Figure 3: Total X-ray (right) and
neutron (left) structure factors, $S_{T}^{x}(k)$ and $S_{T}^{n}(k)$, for
amorphous Na3OCl at 300K, are displayed for varying model sizes. FPMD data are
depicted in orange and grey for the 135 and 405 atoms model, respectively. GAP
data are indicated by light blue, blue, purple, and green for the 135, 405,
1080, and 3645 atoms model, respectively. For $S_{T}^{x}(k)$ and
$S_{T}^{n}(k)$, zoom-in in the 1.5-3.8 Å-1 and 1.4-4 Å-1 ranges are reported,
respectively.

##  III Results and Discussion

In the following sections, we report a detailed analysis of various structural
properties using the results obtained from both FPMD and MLIP schemes. Each
analysis begins with an evaluation of size effects at the FPMD level by
comparing models containing 135 and 405 atoms. Next, we compare the
performance of MLIP against FPMD findings by assessing the congruence between
the structures of the 135 and 405 atom models. Lastly, we extend the size
effect assessment to the MLIP level, incorporating models with up to 3645
atoms.

###  III.1 Total X-ray and Neutron structure factors

The calculated total X-ray and neutron structure factors, denoted as
$S_{T}^{x}(k)$ and $S_{T}^{n}(k)$, for amorphous Na3OCl models are depicted in
Fig. 3. These were obtained through the Fourier transform (FT) of the real-
space pair distribution functions generated by the simulations. While they
exhibit broadly similar profiles, notable differences are observed between the
structure factors of the 405-atom models and those of the 135-atom models,
particularly regarding the peaks’ maximum intensity and positions. For
$S_{T}^{x}$(k), the primary peak at 2.5 Å-1 is more intense and goes along
with a more pronounced shoulder at 1.9 Å-1 in the 405-atom models.
Additionally, the secondary peak, located at 3.2 Å-1 in the 135-atom models,
shifts to a slightly lower $k$-value of 3.1 Å-1 and exhibits a less defined
minimum between the first and second peaks in the larger models. Following the
initial peaks, both the 135-atom and 405-atom models exhibit much broader
maxima and minima and rapidly damped oscillations up to $k\sim$8 Å-1. These
features of the structure factor curves are consistently observed across both
model sizes. In terms of $S_{T}^{n}(k)$, the differences between the 135-atom
and 405-atom models are more pronounced compared to those observed for
$S_{T}^{x}(k)$. Particularly, the 405-atom model shows the absence of the
small peak at $k\sim 1~{}\text{\AA}^{-1}$, indicating a vanishing
intermediate-range order in this amorphous system. Concerning the peaks found
at $k<4~{}\text{\AA}^{-1}$ in the 135 atoms model, the 405-atom model features
a first peak with very similar maximum intensity and position at $\sim
2.0~{}\text{\AA}^{-1}$, whereas the second peak appears at a lower $k$ value
($\sim 2.8~{}\text{\AA}^{-1}$ compared to $\sim 3.0~{}\text{\AA}^{-1}$) and is
more pronounced. The third peak is also shifted to a lower $k$ ($\sim
3.4~{}\text{\AA}^{-1}$ versus $\sim 3.6~{}\text{\AA}^{-1}$) and is slightly
less intense. Additionally, a distinct shoulder at $\sim
3.8~{}\text{\AA}^{-1}$ is observed in the 405 atoms model, which is not
present in the 135 atoms model. The 405-atom FPMD model, currently the largest
FPMD model developed to date, represents our most accurate description of
amorphous Na3OCl. It aligns closely with the reported data from the 135-atom
model, both from this study and previously reported [30], albeit with minor
yet notable differences in peak positions and intensities, denoting minimal
size effects within the size range of 135-atom to 405-atom models.
To further assess the degree of agreement between the FPMD models of 135 and
405 atoms, we employ the $R_{\chi}$ factor [60]. This factor is calculated
using the equation defined in (7), allowing for a quantitative analysis of the
similarity between the calculated $S_{T}^{x}(k)$ and $S_{T}^{n}(k)$ of the two
models and the impact of system size on structural features.

$R_{\chi}=\left(\frac{\sum_{i}~{}\left[S^{\rm fpmd}(k_{i})-S^{\rm
fpmd^{\prime}}(k_{i})\right]^{2}}{\sum_{i}~{}\left[S^{\rm
fpmd}(k_{i})\right]^{2}}\right)^{1/2}$ (7)

In this formula, $S^{\rm fpmd}(k_{i})$ and $S^{\rm fpmd^{\prime}}(k_{i})$
refer to the calculated FPMD structure factor of the 405 and 135 atoms models,
respectively, at a given wave vector $k_{i}$. Table 1 details the calculated
$R_{\chi}$ values for both $S_{T}^{x}(k)$ and $S_{T}^{n}(k)$. Notably, within
glasses simulated using FPMD, $R_{\chi}$ values equal of 10 or lower in the
1-10 $k$ range are commonly accepted as denoting quantitative agreement with
experimental data [bouzid2021]. The computed $R_{\chi}$ values for the
135-atom FPMD models, being 4.7 for X-ray and 7.5 for neutron structure
factors, corroborate the above analyses, showing a reasonably good agreement
with the data from the 405-atom model. However, these values also indicate a
slightly greater discrepancy for $S_{T}^{n}(k)$ compared to $S_{T}^{x}(k)$,
underscoring the nuanced impact of structure factor type on model agreement.
When comparing the total structure factors $S_{T}^{x}(k)$ and $S_{T}^{n}(k)$
for the 135-atom model using GAP and FPMD schemes, remarkably similar outcomes
are observed. The consistency in data between FPMD and GAP is also equally
evident in the 405-atom model for both $S_{T}^{x}(k)$ and $S_{T}^{n}(k)$ total
structure factors. This similarity highlights the effectiveness of the
developed MLIP potential in accurately capturing the structural features of
amorphous Na3OCl, closely mirroring the predictions of the more
computationally accurate and costly FPMD approach. The high degree of
quantitative agreement between GAP and FPMD schemes is further underscored by
the low $R_{\chi}$ values (Table 1). These values, which are below 3 for X-ray
and below 4 for neutron structure factors in the 135-atom models, and less
than 3 for both X-ray and neutron in the 405-atom models, highlight the FPMD
(DFT)-accuracy of the GAP model in replicating the detailed structural
features of amorphous Na3OCl.
Having established the FPMD (DFT) accuracy of our newly developed MLIP GAP
potential, we are in a position to explore size effects by analyzing the 1080
and 3645-atom models obtained with GAP. By comparing these profiles to the
405-atom model via FPMD and GAP, we observe only very minor differences. For
$S_{T}^{x}(k)$, the subtle shoulder at 1.9 Å-1 previously noted in the
405-atom model becomes a discernible, albeit small, peak with increased
intensity in the 1080 and 3645-atom models. Similarly, the intensity of the
first peak maximum in $S_{T}^{n}(k)$ increases in the models with 1080 and
3645 atoms compared to the 405-atom model, highlighting subtle but notable
size-dependent effects.

Figure 4:  Calculated X-ray (right) and neutron (left) total pair correlation functions, $g_{t}^{x}(r)$ and $g_{t}^{n}(r)$, for amorphous Na3OCl models at 300K. FPMD data is reported for 135 (orange) and 405 (grey) atoms models, whereas GAP data is reported for 135 (light blue), 405 (blue), 1080 (purple), and 3645 (green) atoms models.  Table 1: Averaged goodness-of-fit $R_{\chi}$ parameters for the 135 atoms FPMD model and GAP models ranging from 135 to 3645 atoms, indicating their degree of agreement with the reference 405 atoms FPMD model data in terms of neutron and X-ray structure factors ($R_{n}^{S(k)}$ and $R_{x}^{S(k)}$, respectively) and the total pair correlation functions ($R_{n}^{T(r)}$ and $R_{x}^{T(r)}$, respectively). $R_{n},x^{S(k)}$ and $R_{n},x^{T(r)}$ are calculated over the ranges of 0.75-10.0 Å-1 and 0.75-8.0 Å, respectively. For the 135 atoms GAP model, the $R_{\chi}$ values are provided with reference to the 135 atoms FPMD model data, serving for assessing the consistency of the GAP potential across different model sizes. | FPMD | GAP
---|---|---
n. atoms | 135 | 135 | 405 | 1080 | 3645
X-Rays |  |  |  |  |
$R^{\rm x}_{\rm S(k)}$ | 4.7 | 2.7 | 2.6 | 3.5 | 3.6
$R^{\rm x}_{\rm T(r)}$ | 4.9 | 4.8 | 5.4 | 3.6 | 4.3
Neutrons |  |  |  |  |
$R^{\rm n}_{\rm S(k)}$ | 7.5 | 3.9 | 2.6 | 5.4 | 5.3
$R^{\rm n}_{\rm T(r)}$ | 5.8 | 5.7 | 5.3 | 3.9 | 4.3

###  III.2 Total and partial pair-correlation functions

The X-ray and neutron total pair correlation functions,
$g_{\text{tot}}^{x/n}(r)$, for Na3OCl models obtained via FPMD and GAP, are
shown in Fig. 4. These functions are characterized by two primary peaks at
about 2.28 Å and 2.77 Å, with distinct intensities between X-ray and neutron
data. Furthermore, $g_{\text{tot}}^{n}(r)$ exhibits an additional, broader
peak around 4 Å, followzed by reduced oscillations at larger distances. The
presence of distinct peaks in the total pair correlation functions at specific
interatomic distances indicates a consistent short-range order within the
amorphous Na3OCl structure [30]. The additional broader peak observed in
neutron correlation functions implies differences in the way X-ray and neutron
scatterings are sensitive to the specific atomic arrangements.The damped
oscillations in the total pair correlation functions,
$g_{\text{tot}}^{x/n}(r)$, for distances greater than 5 Å lend support to the
previous observations derived from the structure factors $S_{T}(k)$. This
behavior is indicative of a very limited intermediate range order within the
amorphous Na3OCl structure, which aligns with the characteristics typically
associated with amorphous antiperovskites [30]. The close agreement between
the 135 and 405-atom FPMD models implies a negligible size effect on these
correlations at the FPMD level. This is corroborated by the relatively low
$R_{\chi}$ values reported in Table 1, with a $R^{x}_{T(r)}$ of 4.9 and a
$R^{n}_{T(r)}$ of 5.8 for the FPMD 135-atom model when compared to the
405-atom model. The remarkable quantitative agreement between the GAP and FPMD
results is assessed by $R_{\chi}$ values spanning 4.8 to 5.4 for X-rays and
approximately 5.7 to 5.3 for neutrons, for the 135 and 405-atom models,
respectively. The GAP models with 405, 1080, and 3645 atoms are characterized
by highly similar $g_{\text{tot}}^{x/n}(r)$ for both X-ray and neutron,
suggesting a minimal size effect on these correlation functions. However, when
considering the $R_{\chi}$ values, the concordance with FPMD data improves for
the larger models, as shown by the reduced $R_{\chi}$ values, which are close
to $\sim$ 4\. This reflects an enhanced accuracy in structural reproduction as
the GAP model size increases.
Analyzing the partial pair correlation functions gαβ(r) (Fig. 5) allows us to
trace back the role of each individual pair contribution with respect to the
total one. In Tab. 2 we report the nearest-neighbor distances $r_{ij}$
identified by the position of the first maximum of the gαβ(r) for the Na3OCl
models simulated by FPMD and GAP. For comparison, we also report the nearest-
neighbor distances found in synthesized pure crystalline Na3OCl [61, 26].

Figure 5: The partial pair correlation functions $g_{\text{NaO}}(r)$, $g_{\text{NaCl}}(r)$, $g_{\text{NaNa}}(r)$, $g_{\text{OO}}(r)$, $g_{\text{OCl}}(r)$, and $g_{\text{ClCl}}(r)$ for amorphous Na3OCl at $T=300$ K are shown, as obtained comparing 135 and 405 atoms FPMD models (a), 135 atoms FPMD and GAP models (b), and 405 atoms FPMD model with 405 (c), 1080 (d) and 3645 (d) atoms GAP model. The plots are vertically offset for better visibility.  Table 2: Upper part: nearest-neighbour interatomic distances $r_{ij}$ (in Å) identified by the position of the first maximum of the pair correlation functions $g_{ij}(r)$. For comparison, we report the values of the crystalline Na3OCl phase obtained by experiments.[43, 61] The average statistical error for the simulated $r_{ij}$ data (FPMD and GAP), is 0.02 Å. Center part: total and pair coordination numbers as defined by taking different cutoff radii for the definition of a total or a “cation-anion” shell of interactions. Values in parenthesis correspond to the statistical uncertainty on the last reported digits. Lower part: density ($d$, g/cm3) of the models simulated. For GAP models $d$ is computed along 100 ps at 300 K and 1 Katm performed in the NPT ensemble. GAP models data is averaged over 11 replica for 135-atom model and over 2 replica for the other GAP models. | Exp.[43, 61] | FPMD | GAP
---|---|---|---
| Cr. | Am. | Am. | Am. | Am. | Am. | Am.
n. atoms | – | 135 | 405 | 135 | 405 | 1080 | 3645
$r_{\rm NaO}$ | 2.25 | 2.28 | 2.30 | 2.28 | 2.27 | 2.27 | 2.26
$r_{\rm NaCl}$ | 3.18 | 2.77 | 2.75 | 2.75 | 2.76 | 2.78 | 2.79
$r_{\rm NaNa}$ | 3.18 | 3.19 | 3.11 | 3.16 | 3.15 | 3.13 | 3.14
$r_{\rm OO}$ | 4.50 | 3.83 | 3.79 | 3.81 | 3.72 | 3.74 | 3.73
$r_{\rm OCl}$ | 3.89 | 3.90 | 3.94 | 3.95 | 3.95 | 3.95 | 3.93
$r_{\rm ClCl}$ | 4.50 | 3.96 | 3.89 | 4.00 | 3.86 | 3.93 | 3.94
$n^{t}_{\rm Na}$ | 14 | 8.06(4) | 8.16(3) | 8.16(9) | 8.23(2) | 8.24(3) | 8.24(2)
$n^{p}_{\rm Na}$ | 6 | 4.0(1) | 4.10(2) | 4.06(5) | 4.03(3) | 4.09(2) | 4.09(1)
$n^{t}_{\rm O}$ | 6 | 6.40(3) | 6.53(3) | 6.34(8) | 6.44(3) | 6.46 (3) | 6.40(2)
$n^{t}_{\rm Cl}$ | 12 | 5.75(4) | 5.84(4) | 5.85(7) | 5.79(3) | 5.88 (3) | 5.87(2)
$d$ (g/cm3) | 2.20 | 2.04 | 2.04 | 2.01 | 2.02 | 2.01 | 2.02
Error(%) | – | – | – | (-1.3%) | (-1.0%) | (-1.3%) | (-1.0%)

The first peak of $g_{\text{tot}}(r)$ for amorphous Na3OCl FPMD models can be
attributed entirely to the Na-O pair at approximately 2.28-2.30 Å, while the
second peak is primarily due to the Na-Cl pair at roughly 2.75-2.77 Å, with
only partial contributions from Na-O and Na-Na pairs within 3.11-3.19 Å in the
$g_{\alpha\beta}(r)$ functions. The amorphous nature of the network favors
relatively short Na-Cl distances ($\sim$2.75-2.77 Å) when compared to those
found in the crystalline phase, where each Cl atom is surrounded by twelve Na
atoms at a distance of $\sim$3.18 Å. The initial peak of $g_{\text{NaNa}}(r)$
is responsible for the broadened profile of the second peak observed at about
2.8-3.1 Å in the total pair correlation functions. Upon analyzing the peak
positions in $g_{\text{NaO}}(r)$ for amorphous Na3OCl, the Na-O bond distance
is slightly larger than the typical ionic bond seen in crystalline Na3OCl,
which is around 2.25 Å, whereas the Na-Cl ionic bond distance is shorter
compared to the crystalline counterpart at approximately 3.18 Å. The anion-
anion pair correlation functions, $g_{\text{OO}}(r)$, $g_{\text{ClCl}}(r)$,
and $g_{\text{OCl}}(r)$, exhibit a principal peak centered near 3.8-4.0 Å.
Both the 135 and 405 atoms FPMD models shows very similar partial pair
correlation functions $g_{\alpha\beta}(r)$, with the 405 atoms model having a
slightly narrower first peak for the Na-O pair, a slightly shorter Cl-Cl (3.9
Å vs 4.0 Å) and more distinct first peaks for O-O. In the 135 atoms model, the
O-O pair presents a broadened peak centered around 3.83 Å, while in the 405
atoms model, it displays a primary peak at approximately 3.79 Å with an
additional shoulder at around 4.1 Å. Notably, the O-O pair interaction is of
crucial significance, as the interaction between O atoms has been identified
as critical for the electronic and electrochemical stability of amorphous
Li3OCl in contact with a Li metal electrode [62].
The GAP models accurately reproduce the $g_{\alpha\beta}(r)$ functions
observed in both the 135 and 405 atoms FPMD models, further validating the
FPMD (DFT)-level accuracy of the GAP approach developed for amorphous Na3OCl.
Notably, the discrepancies between the 135 and 405 atoms FPMD models are
precisely captured by the GAP models, confirming that these variations can be
ascribed to size effects. The GAP calculations for models containing 405,
1080, and 3645 atoms yield remarkably consistent results across all pairs,
with the exception of the $g_{\text{OO}}(r)$ function’s first peak, which
appears slightly more pronounced.
In Table 2, we include the density ($d$, in g/cm3) of the amorphous Na3OCl
models simulated by FPMD and GAP. We remind that FPMD carried in the NVT
ensemble were performed at a equilibrated reduced density of about 7% relative
to the crystalline phase (corresponding to a volume expansion of +5%),
consistent with our previous work [30]. This adjustment minimized the stress
tensor to values closed to 0 GPa for an energy cutoff of 160 Ry for the plane
wave expansion. To further evaluate the performance of the GAP potential
concerning density, we conducted MD simulations in the NPT ensemble at 300 K
and 1 atm, averaging the final models’ volume over 100 ps. The resulting error
was less than 1.5%, further confirming the reliability of the developed GAP
potential.

Figure 6: Distributions of the structural units for Na of amorphous Na3OCl as a function of the number of neighbors in each unit ($l$). Values obtained by using the definition of “total” coordination $n^{t}_{ij}$ , given in Section IVC. Table 3: Distribution of the individual cation-anion n${}_{\alpha}(l)$ structural units where an atom of species $\alpha$ (Na, Cl or O) is $l$-fold coordinated to a counter ion computed for glassy Na3OCl (see definition 1, Sec. “Total and partial coordination numbers”). In bold are reported the total percentages determined for each $l$-fold coordination. These quantities have been calculated including neighbours separated by a cutoff corresponding to the first minimum in the $g_{\alpha\beta}(r)$. For the present work, the individual pair cutoffs used are 3.30, 3.94, 4.10, 5.06, 5.51, and 5.39 Å for, respectively, the Na-O, Na-Cl, Na-Na, O-O, Cl-Cl and O-Cl distances. A total cutoff of 3.30 Å was defined from the total $g_{\alpha\beta}(r)$. Only fraction greater than 1.5 % are reported. Error bars are given in parenthesis.  |  | Proportions n̄${}_{\alpha}(l)$ (%)
---|---|---
|  | FPMD | GAP
n. atoms | 135 | 405 | 135 | 405 | 1080 | 3645
Na |  |  |  |  |  |  |
$l$=3 |  |  15.1(0.8) |  14.7(0.8) |  10.9(1.5) |  14.4(0.9) |  14.4(0.6) |  14.3(0.3)
| O2Cl1 | 10.5 | 7.3 | 6.9 | 8.8 | 8.7 | 7.8
| O3 | 3.9 | 6.3 | 3.4 | 4.5 | 4.6 | 5.3
$l$=4 |  |  66.4(3.2) |  62.1(0.9) |  72.9(2.6) |  67.8(2.0) |  63.2(1.1) |  59.90(0.6)
| O2Cl2 | 35.9 | 28.2 | 36.5 | 34.2 | 31.4 | 29.5
| O3Cl1 | 15.1 | 18.2 | 19.9 | 16.9 | 16.7 | 14.4
| O1Cl3 | 10.2 | 10.3 | 14.2 | 12.1 | 11.0 | 12.0
| O4 | 4.3 | 4.7 | 2.1 | 3.9 | 4.0 | 3.9
$l$=5 |  |  16.6(2.8) |  19.7(0.8) |  14.9(1.8) |  17.0(0.8) |  20.0(0.7) |  17.5(0.4)
| O2Cl3 | 6.5 | 7.1 | 7.0 | 6.7 | 8.5 | 8.3
| O1Cl4 | 5.3 | 7.9 | 5.1 | 5.7 | 6.7 | 5.1
| O3Cl2 | 1.7 | 3.4 | 1.8 | 2.8 | 3.4 | 3.0
$l$=6 |  |  2.1(1.2) |  2.1(0.2) |  1.1(0.4) |  0.6(0.2) |  2.2(0.3) |  1.7(0.3)
O |  |  |  |  |  |  |
$l$=5 | Na5 |  1.2(0.9) |  0.7(0.5) |  1.1(0.8) |  1.0(0.9) |  1.0(0.5) |  1.1(0.3)
$l$=6 | Na6 |  66.8(3.3) |  53.6(0.9) |  71.3(1.3) |  67.7(2.1) |  61.5(1.8) |  62.2(0.7)
$l$=7 | Na7 |  30.0(3.3) |  43.21(1.1) |  24.8(1.9) |  27.7(2.4) |  34.2(1.9) |  32.6(1.2)
$l$=8 | Na8 |  1.5(1) |  2.4(1.4) |  2.8(1.1) |  2.6(1.4) |  3.3(1.0) |  4.0(0.6)
Cl |  |  |  |  |  |  |
$l$=4 | Na4 |  7.4(1.4) |  3.7(0.9) |  3.9(1.9) |  6.2(1.8) |  6.1(1.3) |  5.9(0.8)
$l$=5 | Na5 |  31.0(1.6) |  30.5(1.0) |  28.5(2.3) |  31.9(2.2) |  27.8(1.8) |  28.0(1.4)
$l$=6 | Na6 |  41.7(2.6) |  38.5(0.9) |  45.2(1.9) |  42.3(2.3) |  41.0(2.1) |  41.4(1.7)
$l$=7 | Na7 |  16.9(1.2) |  19.7(0.9) |  19.4(1.8) |  17.3(1.9) |  20.4(1.5) |  20.0(1.1)
$l$=8 | Na8 |  2.0(0.9) |  3.9(0.5) |  2.6(1.6) |  2.0(1.4) |  4.1(1.1) |  3.9(0.6)

###  III.3 Total and partial coordination numbers

Further understanding of the Na3OCl network is garnered through the analysis
of total and partial coordination numbers ($n_{i}$ and $n_{ij}$,
respectively), derived from the pair correlation functions. Specifically,
$n_{ij}$ is calculated by integrating the first peak of $g_{ij}(r)$ up to the
cutoff distance defined by the first minimum. The total coordination numbers
for Na, O, and Cl are computed as follows: $n_{Na}=n_{NaO}+n_{NaCl}+n_{NaNa}$,
$n_{O}=n_{ONa}+n_{OCl}+n_{OO}$, and $n_{Cl}=n_{ClNa}+n_{OCl}+n_{ClCl}$. A
clear minimum following the first peak of $g_{ij}(r)$ allows for an
unambiguous definition of $n_{ij}$, as the running coordination number shows a
distinct plateau (e.g., Na-O pair [30]). However, in the absence of this
feature (e.g., Na-Na pair [30]), $n_{ij}$ becomes more sensitive to the cutoff
choice. To address this issue, we use individual pair cutoffs determined by
the minima following the first peak of $g_{ij}(r)$ and a ”total”
cutoff—averaged across three amorphous models at 3.3 Å—following the second
peak of $g_{\text{tot}}(r)$, as in previous work [30]. This approach considers
all counter-ions (O and Cl) and some Na neighbors within Na’s first
coordination shell, facilitating a comparison with crystalline Na3OCl
coordination numbers. Crystalline Na3OCl is marked by Na atoms coordinated by
six counter-ions (two O and four Cl) within the nearest-neighbor shell, with O
atoms coordinated by six Na atoms and Cl atoms by twelve Na atoms. With our
chosen cutoff (3.3 Å), each Na atom also neighbors eight Na atoms, leading to
a ”total” Na coordination of 14 [43, 61]. This crystalline Na environment
comprises 14 neighboring atoms, linked via Na-O-Na connections [30].
Accordingly, we analyzed amorphous Na3OCl models’ coordination numbers
$n_{ij}$ using ”partial” $n^{p}_{ij}$ (for cation-anion interactions,
definition 2) and ”total” $n^{t}_{ij}$ coordination numbers (for a
comprehensive neighbor overview, definition 1). This distinction helps
characterizing each atom’s surrounding structural units, revealing the network
organization and providing insights into the specific chemical bonding nature
going beyond average behaviors. We present the calculated total coordination
numbers ($n^{t}_{\text{ij}}$) for the three atomic species and the partial
coordination numbers ($n^{p}_{\text{ij}}$) for Na in Table 2.
Focusing on the structural units, Fig. 6 illustrates their distribution based
on the number of neighbors ($l$) for Na, utilizing definition 2. For Na atoms
within a 3.3 Å cutoff, amorphous Na3OCl shows a significantly different
behavior compared to its crystalline counterpart. Specifically, the
distribution of Na atom units spans a wide range (from 4 to 11) and is
centered around $l=8.1$-$8.2$, with the count of neighboring counter-ions and
Na neighbors diminishing from 6 and 8 to about 4.0-4.1, respectively. The FPMD
and GAP models exhibit highly consistent results across all model sizes,
within an averaged uncertainty bar of ±0.1 for coordination numbers. This
consistency is indicative of a minimal size effect on this specific property.
By using definition 1, we report the distributions of structural units for Na,
O, and Cl, focusing on interactions with what we designated as counter-ions.
These distributions are given along with the detailed chemical composition
breakdown of each structural unit for a specific $l$.

###  III.4 Chemical identification of structural units

The coordination units identified in the various FPMD and GAP models of
amorphous Na3OCl are detailed in Table 3. At the FPMD level, both the 135 and
405 atoms models exhibit similar trends, suggesting a predominantly fourfold
first coordination shell for Na ($\sim$66% and 62%, respectively) with respect
to cation-anion (counterion) interaction. The breakdown by chemical species
for the 135 atoms model aligns closely with our prior findings [30],
showcasing, for decreasing percentages, the units O2Cl2 ($\sim$35.9%), O3Cl1
($\sim$15.1%), O1Cl3 ($\sim$10.2%), and O4 ($\sim$4.3%). The newer 405 atoms
FPMD model has a slightly different distribution, with O2Cl2 at $\sim$28.2%
and O3Cl1 at $\sim$18.2%, while O1Cl3 and O4 figures align with previous
observations. Threefold and fivefold units amounts to about $\sim$15% and
$\sim$17-20% respectively, with a notable occurrence of structures having two
or one oxygen atom (O2Cl1, O2Cl3, and O1Cl4). The results from the 135 and 405
atoms models are highly consistent within the statistical error, indicating
reduced variability for the larger model compared to the smaller one. Our
findings reveal that the structure of amorphous Na3OCl markedly deviates from
the molecular dynamics model proposed for glassy Li3OCl, which suggests
nanophase segregation into regions predominantly composed of Li2O and LiCl
[63]. Specifically, when examining Na atoms that are fourfold or threefold
coordinated in amorphous Na3OCl, only approximately 8-11%, in total, are
solely coordinated with O atoms (O3 and O4), and less than 1.5% are
exclusively coordinated with Cl atoms, fully in line with our previous finding
[30].
For oxygen and chlorine, we focus solely on the analysis based on the total
coordination number ($n^{t}_{\text{O/Cl}}$). Consistently with our previous
findings [30], their local environments are predominantly made of counter-ions
(Na), contrasting with the local environment of Na atoms. The value of
$n^{t}_{\rm O}$ in amorphous Na3OCl is about $\sim$6.4-6.5 (Table 2), close to
the one in crystalline Na3OCl [43, 61, 26]. We observe that about 67% of O
atoms are coordinated by six Na atoms, and about 30% by seven Na atoms in the
135 atoms FPMD model. There are notably different outcomes for the 405 atoms
FPMD model, with about 54% and 43% of O atoms exhibiting six- and seven-fold
coordination with Na, respectively (Table 3). Despite these variations in
distribution units, the difference in the total coordination number for O
($n^{t}_{\text{O}}$) between the two model sizes is minor (6.4 vs. 6.5). For
Cl, $n^{t}_{\rm Cl}$ is equal to $\sim$5.8 for both FPMD sizes, almost half of
that found in the crystalline phase (12). Several structural units connected
to Cl are noticeable in Table 3. Indeed, approximately 31% of Cl atoms in our
simulations are coordinated by six Na atoms, while about 42-39% are surrounded
by five Na atoms, and roughly 17-20% by seven Na atoms. The proportion of
four- and eight-fold coordinated Cl atoms changes when comparing the 135 and
405 atoms models. Specifically, the 135 atoms model has about 7% four-fold and
2% eight-fold coordinated Cl atoms, whereas the 405 atoms model exhibits a
lower presence of four-fold coordinated Cl (4%) and a doubled content of
eight-fold coordination (around 4%).
For the GAP case, the 135 and 405 atoms models exhibit total coordination
numbers for the three species (Na, O, and Cl) that are closely aligned with
those derived from FPMD simulations, with values equal to 8.2, 6.3-6.5, and
5.8-5.9, respectively. However, a closer look to the individual structural
unit distributions reveals that both GAP models feature distributions similar
to those of the FPMD model for Na and Cl, within statistical uncertainty. In
contrast, for O, the distributions in both GAP models more closely match the
135 atoms FPMD result. Within the GAP scheme, increasing the model size to
1080 and 3645 atoms results in a structural unit distributions of O atoms
approaching values similar to those observed in the 405 atoms FPMD model.
Overall, the comprehensive analysis of amorphous Na3OCl models through both
FPMD and GAP methodologies has yielded significant insights into the
structural characteristics of this material. The total and partial
coordination numbers for Na, O, and Cl have revealed the subleties of the
local atomic environments in amorphous Na3OCl. The predominant coordination
patterns for Na indicate a mix of threefold, fourfold, and higher coordination
states, reflecting a complex network structure that diverges from the simpler
models conjectured for glassy Li3OCl. Both FPMD and GAP schemes exhibit
remarkable consistency in the calculated total coordination numbers for all
atomic species across different model sizes, confirming the occurrence pf
minimal size effects. The distributions of structural units further emphasize
the accuracy of GAP models in reproducing FPMD-derived structures,
underscoring the potential of GAP as an efficient and reliable tool for
studying amorphous antiperovskite materials. The analysis focused on O and Cl
reveals that their local environments are predominantly made up of counter-
ions (Na), a finding substantiating our previous study. This consistency
across different models and sizes for O and Cl reveal the structural stability
and uniformity of the amorphous Na3OCl network. The detailed characterization
of amorphous Na3OCl provides valuable insights into its potential applications
and stability. Understanding the coordination environments and the
distribution of structural units is crucial for predicting material
properties, such as ionic conductivity and stability, which are essential for
applications in energy storage and conversion technologies and which will be
focus of future effort.

##  IV Conclusions

The detailed study of amorphous Na3OCl through FPMD and MLIP simulations has
provided profound insights into its atomic structure and coordination
environment. Our findings reveal a complex network structure with varying
coordination patterns, significantly differing from the simpler conjectured
models for glassy Li3OCl. The consistency in total coordination numbers across
different model sizes and methodologies highlights a negligible size effect on
the material’s structural properties. The GAP model, in particular,
demonstrates remarkable fidelity in capturing the structural features of the
amorphous Na3OCl, as assessed by FPMD simulations, across an extended range of
model sizes up to 3645 atoms. The analysis of oxygen and chlorine environments
underscores their predominantly counter-ion composed local environments,
differing from Na’s local environment. This study not only advances our
understanding of amorphous Na3OCl’s structural characteristics but also
showcases the effectiveness of combining FPMD and MLIP approaches for
electrolyte materials’ analysis. Our findings lay a groundwork for future
studies into the properties of amorphous materials and their potential
technological applications, particularly in the field of energy storage and
conversion technologies.

## DATA AVAILABILITY

The GAP potential, is freely available at YYY for non-commercial research.
Representative configurations of Na3OCl FPMD and MLIP models and raw
trajectory data are available at the European Center of Excellence Novel
Materials Discovery (CoE-NOMAD) repository (see ref. [64]).

###### Acknowledgements.

This work was supported by the National Research Foundation of Korea (NRF)
grants funded by the Ministry of Science and ICT (MSIT), government of South
Korea (grant no. 2021R1F1A1048555). We acknowledge financial support from the
Agence Nationale de la Recherche (ANR) within the framework of the project
AMSES (ANR-20-CE08-0021) and the EUR-QMAT-ANR under the contract
ANR-11-LABX-0058-NIE and ANR-17-EURE-0024 within the Investissement d’Avenir
program ANR-10-IDEX-0002-02. We also acknowledge support from the Seed Money
program (project MEDIA) of Eucor-The European Campus. This project has also
received funding from the European Union’s Horizon 2020 research and
innovation programme under the Marie Skłodowska-Curie grant agreement number
847471 (QUSTEC). Calculations were performed by using resources from GENCI
(Grand Equipement National de Calcul Intensif, grant no. A0120807670,
A0140807670, and A0140912441) and the Pole HPC Equipe@Meso of the Université
de Strasbourg.

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Heliophysics Decadal Survey 2024 White Paper Lunar Samples are Time Capsules
of the Sun

Authors: Prabal Saxena* (NASA Goddard), Natalie Curran (Catholic
University/CRESST II/NASA Goddard), and Heather Graham (NASA Goddard)

Co-Signers: Vladimir Airapetian (American University/NASA Goddard), D.L. Blank
(University of Southern Queensland), Ian Crawford (Birbeck, University of
London), Ben Davidson (SpaceWeatherNews, The Mobile Observatory Project),
Brett W. Denevi (Johns Hopkins University Applied Physics Laboratory),
Katherine Joy (University of Manchester), James Tuttle Keane (NASA JPL),
Rosemary M. Killen (NASA Goddard), Jason L McLain (NASA Goddard), Liam
Morrissey (Memorial University), Noah Petro (NASA Goddard), Carle M. Pieters
(Brown University), Song Tan [谭宋] (Yunnan Observatories, Chinese Academy of
Sciences)

Synopsis/Recommendation: The history of the Sun is buried in the surface of
the Moon, which presents a particularly exciting opportunity given the
upcoming planned efforts for the return of humans to the Moon. These future
efforts involve long term, sustainable human exploration of the Moon and
promise a return of a large mass of diverse and new types of lunar samples.
The Heliophysics Decadal survey should actively embrace this coming
opportunity and facilitate cross-disciplinary efforts to unlock the secrets of
the Sun held by the lunar surface. With planned Artemis efforts that include
prioritization of samples of high interest and protocols for sample handling
and analysis, input into relevant solar signatures that would be most
diagnostic and how best to obtain/retain them is incredibly important.
Finally, leveraging the theoretical expertise of the two communities in ways
that bring them together, such as through dedicated conferences and workshops,
will let the two communities help each other learn more than they could alone.
*Contact Email<EMAIL_ADDRESS>
The history of the Sun is buried in the surface of the Moon. This is in part
due to the profound effect the Sun has on bodies in the solar system [29, 19]
\- one recognized as a key goal of the previous heliophysics decadal survey
("Determine the interaction of the Sun with the solar system…") [8]. While the
Sun sculpts the atmospheres, surfaces and habitability of the larger
atmosphere possessing bodies in our solar system, its interactions with the
Moon’s surface are more direct as they have been potentially mediated by an
atmosphere or a global magnetic field for only brief periods of time [26, 21,
34, 30, 13]. Consequently, interactions of both the energetic and more
corpuscular (magnetized solar wind and energetic particles) outputs of the Sun
are able to leave seemingly indelible signatures in the lunar surface.

Figure 1: Examples of potential paleoregolith sites on the Moon and the
process by which paleoregoliths are sealed off and may provide a valuable
record of past solar interactions with the lunar surface (from [9]).

A key advantage of signatures of the Sun left on the lunar surface is that
there are very few processes (e.g. plate tectonics driven resurfacing,
erosion, etc.) which have operated globally on the Moon and which could have
wiped out such signatures. Processes such as volcanism and impacts have
modified parts of the surface, but have also done so in a way which has been
able to leave many key chronologically diagnostic signatures intact. Indeed in
both cases, these process may actually serve to help highlight and constrain
the chronological influence of the Sun in specific periods of time. For
example, in the case of volcanism, lava flows from episodic volcanism may have
essentially sealed off layers of the regolith from further interaction with
the Sun [14] that leaves these overlaid portions of ‘paleoregolith’ as
invaluable snapshots into time periods preceding the event in which they were
trapped (see figure 1 for an example from [9]). Similarly, while impacts of
high enough energy may in some cases modify or eliminate signatures of
interest in specific regions, the impact process also drives the overturning
and stratification of the regolith [17, 7] \- a process which creates a
chronological record of surface interactions with the Sun with depth.

The existence of potential chronologically diagnostic signatures of solar
interaction with the lunar surface, particularly paleo-space weather, is
especially exciting given the demonstrated ability to repeatedly actively
obtain samples from the Moon. Samples from the lunar surface have been
returned to the Earth by multiple space agencies [28, 32], beginning roughly a
half century ago, and continue to this day [5]. This is true for no other body
outside of Earth in the solar system, and as discussed above, lunar samples
have the advantage of preserving a relatively intact record of interaction
with solar and solar system processes in the past. The ability of lunar
samples to explore the nature of the Sun has been recognized by scientists
since the Apollo program. In 1979, a conference on ‘The Ancient Sun: Fossil
Record in the Earth, Moon, and Meteorites’ [1] included seminal initial
presentations on topics including the variation in past solar flare activity,
properties of the past solar wind, changes in the solar dynamo over time and
changes in GCR flux as modulated by the Sun - all deduced from examination of
lunar samples [10, 23, 15, 6, 24]. Additional studies including some recent
ones built on this work by noting the relative constancy of the solar wind in
the recent past (up to 10’s of millions of years) [25, 11, 22], but a
significant number of studies, have noted evidence that the solar wind flux
and consequently the Sun has not been constant farther in the past [23, 31, 4,
12].

Table 1: Selected Lunar Sample Proxies and Relevant Solar Inputs

Proxy | Meaning | Solar Assumptions | Solar Input
---|---|---|---
| Cosmic Ray
---
Exposure Age
| Indicates exposure to cosmic rays
---
(down to several meters)
| Assumes a constant
---
flux of GCRs and
SEPs over time
| SEP/GCR Flux/Spectrum over time,
---
Heliosphere properties that
modulate GCR flux/spectra
| Particle/Fission
---
Tracks
| Indicates irradiation of a
---
particular sample over an
integrated time by
energetic particles
| Varying assumptions about
---
the flux and plausible
spectra of input GCRs/SEPs
| SEP/GCR Flux/Spectrum over time,
---
Heliosphere properties that
modulate GCR flux/spectra
Antiquity Age |  | Semi-quantifies formation age
---
of a breccia - the time at which a
soil is lithified into a breccia
| Dependent on isotope ratios
---
normalized to a constant
solar wind flux
| Overall SEP flux rate from Solar
---
Wind (particularly Argon),
Solar UV flux input for ionization
Maturity Index |  | Indicates surface exposure
---
of particular sample
(in top mm of the surface)
| Dependent on assumption of
---
constant solar wind particle flux
| Solar wind flux rate - isotopes
---
of helium-3, neon-20 and argon-36

Recent changes in understanding on how the Sun may have evolved over time have
supported this past variable history and have complicated the ability to use
long standing proxies used to interpret the history of and nature of lunar
samples. Many of these proxies (a selection of which are listed in table 1)
are dependent upon assumptions that recent research suggests are untrue, such
as the assumption of constancy of energetic particle flux from the Sun over
time or the constancy of the morphology of the heliosphere over time. These
processes are dependent on fundamental processes and properties of the Sun,
such as rotation rate, and a number of recent studies have suggested that
relaxing such assumptions leads to dramatic impacts on the atmospheres and
surfaces of planets in our solar system [2, 18] and would obviously also
change some interpretation of certain lunar samples using the proxies in table
1.

Figure 2: (Left) A plot showing potential rotation evolution scenarios for the
Sun based upon solar analogue observations [16]. The different potential
rotation evolution pathways translate to significantly different solar
activity and GCR flux. (Right) A cartoon image of the heliosphere and
different high energy particle processes relevant to estimates of cosmic ray
exposure ages [20] – the changing rotation of the Sun would also influence
heliosphere morphology.

Equally intriguing is that the dependence of these proxies used to examine
lunar samples on these Sun related properties may enable the use of lunar
samples to put constraints on the properties of the Sun over time. For
example, examination of solar analogue stars has suggested that the Sun may
have had a range of evolutionary paths with respect to how its rotation
changed over time (see figure 2 \- with significant consequences for the
planets in our solar system [16, 3]. The dependence of the activity of the Sun
and nature of the heliosphere on these different rotation evolution paths
could lead to significantly different signatures in lunar samples,
particularly if there is chronological granularity provided by samples.
Indeed, initial work has attempted to place constraints on the likelihood the
Sun had particular rotation states at much younger ages [27].

The potential for lunar samples to hold diagnostic information regarding the
nature of the Sun is especially promising giving planned return of humans to
the Moon. Many of these plans stress the importance of understanding the Sun
over time using lunar samples [33]. These future efforts involve long term,
sustainable human exploration of the Moon and promise a return of a large mass
of diverse and new types of lunar samples. This windfall of information
promises to offer insight into a number of different solar system processes,
and particularly relevant to studies regarding the evolution of the Sun, may
enable population studies that put constraints on these processes over time.
The interpretation of such signatures will require the joint expertise and
efforts of planetary scientists and heliophysicists.

The Heliophysics Decadal survey should actively embrace this coming
opportunity and facilitate cross-disciplinary efforts to unlock the secrets of
the Sun held by the lunar surface. With planned Artemis efforts that include
prioritization of samples of high interest and protocols for sample handling
and analysis, input into relevant solar signatures that would be most
diagnostic and how best to obtain/retain them is incredibly important.
Finally, leveraging the theoretical expertise of the two communities in ways
that bring them together, such as through dedicated conferences and workshops,
will let the two communities help each other learn more than they could alone.

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familyname_firstname

11832 DESY-PROC-2015-02 Patras 2015

# Light Collection in the Prototypes of the ANAIS Dark Matter Project

J. Amaré1,2 , S. Cebrián1,2 , C. Cuesta1,2111Present address: CENPA and
Department of Physics, University of Washington, Seattle, WA, USA , E.
García1,2 , M. Martínez1,2222Present address: Univ. Roma La Sapienza, Roma,
Italy , M.A. Oliván1,2333Corresponding author (e-mail: maolivan@unizar.es), Y.
Ortigoza1,2 , A. Ortiz de Solórzano1,2 , C. Pobes1,2444Present address:
Instituto de Ciencia de Materiales de Aragón, CSIC - Universidad de Zaragoza,
Spain , J. Puimedón1,2 , M.L. Sarsa1,2 , J.A. Villar1,2 , P. Villar1,2
1Laboratorio de Física Nuclear y Astropartículas  Universidad de Zaragoza
Pedro Cerbuna 12  50009  Zaragoza  Spain
2Laboratorio Subterráneo de Canfranc  Paseo de los Ayerbe s/n  22880 Canfranc
Estación  Huesca  Spain

###### Abstract

The ANAIS experiment aims at the confirmation of the DAMA/LIBRA signal using
the same target and technique at the Canfranc Underground Laboratory (LSC) in
Spain. ANAIS detectors consist of large NaI crystals coupled to two
photomultipliers (PMTs). In this work we present Single Electron Response
(SER) data for several units of the Hamamatsu R12669SEL2 PMT model extracted
from normal operation data of ANAIS underground prototypes and we compare them
with PMT SER characterization previously done at surface lab before coupling
them to NaI crystal. Moreover, total light collection for different ANAIS
prototypes has been calculated, producing an excellent average result of 15
phe/keV, which has a good impact in both energy resolution and threshold.

##  1 Introduction

The ANAIS (Annual Modulation with NaI(Tl) Scintillators) experiment [1, 2] is
intended to confirm the DAMA/LIBRA signal [3] using the same target and
technique at the Canfranc Underground Laboratory. The ANAIS-25 set-up
consisted of two NaI(Tl) detectors of 12.5 kg each manufactured by Alpha
Spectra (named D0 and D1 in this work). It has been taking data since December
2012 in order to measure the internal contamination of the NaI(Tl) crystals
and assess the performance of the detectors. A new Alpha Spectra detector
(named D2 in this work) with lower internal background [4] was received in
March 2015, which added to ANAIS-25 modules formed the ANAIS-37 set-up. Every
detector has been coupled to two Hamamatsu R12669SEL2 PMTs, the ANAIS selected
model [5]. In the following we will report on the PMT Single Electron Response
(SER) data extracted from both set-ups on underground site and along normal
operation which have been compared with the SER characterization of the PMTs
previously performed at Zaragoza (Section 2) and on the estimates of the total
light collection for all the available detectors (Sections 3 and 4).

##  2 SER extraction

First, the PMTs SER was measured at the Zaragoza test bench using UV LED
illumination of very low intensity, and triggering in the excitation LED
signal. This characterization was done for each PMT unit before mounting ANAIS
detectors, and allowed to validate the SER determination onsite along normal
operation of ANAIS detectors at the LSC. The latter was performed applying a
peak identification algorithm to the data taken at the LSC, building the SER
by selecting peaks at the end of the pulse of each PMT in order to avoid
trigger bias in pulses having a low number of peaks to prevent the pile-up of
several photoelectrons (phe). An example of a pulse fulfilling these
conditions can be seen in Figure 1(a) and the mean pulse of a selection of
this kind of events is shown in Figure 1(b). The phe area (proportional to
charge) is integrated in a fixed time window around the peak maximum in order
to obtain the single electron response charge distribution. The SER charge
distribution extracted for the same PMT by these two methods is compared in
Figure 1(c) showing full agreement between both.

(a)

(b)

(c)

Figure 1: Pulse with a low number of phe; peaks identified by the applied
algorithm are shown with red triangles (a), SER mean pulse (b) and SER charge
distributions derived at PMT test bench (red) and along normal operation
(blue) (c).

##  3 ANAIS-25

The light collected by each of the PMTs coupled to the ANAIS-25 modules was
calculated by dividing the mean value of the charge distribution associated to
a known energy deposition in the NaI crystal and the mean value of the SER
charge distribution derived as aforementioned. The 22.6 keV line from a
${}^{109}Cd$ calibration source was used for this study. The result of the SER
charge spectrum and the ${}^{109}Cd$ line Gaussian fits can be seen in Table 1
(PMT ij corresponds to PMT j of detector Di). These results and the global
light collection of the two ANAIS-25 detectors are summarized in Table 3. The
${}^{109}Cd$ line resolution is also calculated and is shown in Table 3. These
results confirm the prototypes outstanding light collection and its impact in
resolution. The very good optical performance of the Alpha Spectra modules
evidenced by these figures is very promising in order to reduce the energy
threshold below 2 keVee [6].

PMT |  | SER mean
---
mV·ns
| SER $\sigma$
---
mV·ns
| 22.6 keV mean
---
mV·ns
| 22.6 keV $\sigma$
---
mV·ns
00 | $35.47\pm 0.35$ | $21.73\pm 0.25$ | $6122\pm 2$ | $669\pm 2$
01 | $29.42\pm 0.21$ | $18.81\pm 0.24$ | $5057\pm 2$ | $568\pm 2$
10 | $41.20\pm 0.25$ | $28.30\pm 0.21$ | $7139\pm 4$ | $809\pm 4$
11 | $44.52\pm 0.29$ | $24.36\pm 0.24$ | $7570\pm 4$ | $825\pm 3$
Table 1: ANAIS-25 values for SER charge distribution and ${}^{109}Cd$ 22.6 keV
line Gaussian fits.

PMT |  | PMT
---
phe/keV
| Detector
---
phe/keV
00 | $7.64\pm 0.08$ | $15.24\pm 0.09$
01 | $7.61\pm 0.05$
10 | $7.67\pm 0.05$ | $15.19\pm 0.07$
11 | $7.52\pm 0.05$
Table 2: ANAIS-25 light collection.

PMT |  | PMT
---
$\sigma$/E (%)
| Detector
---
$\sigma$/E (%)
00 | $10.93\pm 0.03$ | $8.51\pm 0.03$
01 | $11.24\pm 0.04$
10 | $11.33\pm 0.05$ | $8.59\pm 0.04$
11 | $10.90\pm 0.05$
Table 3: ANAIS-25 resolution at 22.6 keV.

##  4 ANAIS-37

The same procedure was repeated with ANAIS-37 setup data. In this setup the
operating voltages of the D0 and D1 detectors were increased in order to
better study the low energy region and for this reason the SER values are
higher. The voltage of the new detector (D2) was selected to have a $10^{6}$
gain value in both PMTs in order to explore the high energy region [6]. The
results of the SER charge distribution and the ${}^{109}Cd$ 22.6 keV line
Gaussian fits can be seen in Table 4. The light collection for every PMT and
detector can be observed in Table 6. The newly extracted values for D0 and D1
are compatible with those obtained for ANAIS-25 (see previous section). Good
values for the new D2 ($\sim 16$ phe/keV) have also been measured having again
a good impact in terms of energy threshold and resolution, crucial for the
sensitivity to WIMPs annual modulation.

PMT |  | SER mean
---
mV·ns
| SER $\sigma$
---
mV·ns
| 22.6 keV mean
---
mV·ns
| 22.6 keV $\sigma$
---
mV·ns
00 | $61.47\pm 0.36$ | $35.02\pm 0.32$ | $10257\pm 5$ | $1126\pm 4$
01 | $58.40\pm 0.71$ | $43.06\pm 0.51$ | $10425\pm 5$ | $1166\pm 4$
10 | $83.24\pm 0.55$ | $46.52\pm 0.51$ | $12820\pm 5$ | $1463\pm 4$
11 | $73.91\pm 0.74$ | $42.04\pm 0.52$ | $12740\pm 5$ | $1404\pm 4$
20 | $42.70\pm 2.10$ | $25.42\pm 1.79$ | $7928\pm 5$ | $909\pm 6$
21 | $44.57\pm 2.10$ | $26.67\pm 1.95$ | $8155\pm 6$ | $930\pm 6$
Table 4: ANAIS-37 values from SER charge distribution and ${}^{109}Cd$ 22.6
keV line Gaussian fits.

PMT |  | PMT
---
phe/keV
| Detector
---
phe/keV
00 | $7.38\pm 0.04$ | $15.26\pm 0.10$
01 | $7.88\pm 0.09$
10 | $6.81\pm 0.05$ | $14.44\pm 0.09$
11 | $7.62\pm 0.08$
20 | $8.21\pm 0.40$ | $16.31\pm 0.56$
21 | $8.09\pm 0.38$
Table 5: ANAIS-37 light collection.

PMT |  | PMT
---
$\sigma$/E (%)
| Detector
---
$\sigma$/E (%)
00 | $10.97\pm 0.04$ | $8.73\pm 0.03$
01 | $11.18\pm 0.04$
10 | $11.40\pm 0.03$ | $8.80\pm 0.03$
11 | $11.02\pm 0.03$
20 | $11.46\pm 0.07$ | $8.99\pm 0.05$
21 | $11.40\pm 0.08$
Table 6: ANAIS-37 resolution at 22.6 keV.

##  5 Conclusion

The PMTs single electron response was characterized along detectors normal
operation and compared with the previous PMTs measurements showing a full
compatibility among them. Using this extraction, an excellent light collection
for the three ANAIS detectors, of the order of $\sim$15 phe/keV, has been
measured. Thanks to this, an energy threshold for the ANAIS experiment at 1
keVee is at reach, depending now on improving the filtering protocols for PMT
origin coincident events, which would significantly improve the sensitivity of
the ANAIS Project in the search for the annual modulation effect in the WIMPs
signal [6].

##  6 Acknowledgments

This work was supported by the Spanish Ministerio de Economía y Competitividad
and the European Regional Development Fund (MINECO-FEDER) (FPA2014-55986), the
Consolider-Ingenio 2010 Programme under grants MULTIDARK CSD2009-00064 and
CPAN CSD2007-00042, and the Gobierno de Aragón (GIFNA and ARAID Foundation).
P. Villar is supported by the MINECO Subprograma de Formación de Personal
Investigador. We also acknowledge LSC and GIFNA staff for their support.

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