# Algorithmic detection of the crystal structures from three-dimensional real-
space analysis

Sumitava Kundu  Kaustav Chakraborty  Avisek Das<EMAIL_ADDRESS>School of
Chemical Sciences, Indian Association for the Cultivation of Science, Kolkata,
India, 700032

###### Abstract

Crystal structure detection from the real-space analysis only is still a big
challenge in multiple disciplines due to the absence of exact analytical or
computational methods. New types of complex crystal structures seek more
involvement of advanced techniques and understandings as well as developing
new kinds of machinery. Despite the existence of proper experimental
techniques of crystal structure detection, it lacks the exact approach to
obtain the complete unit cell information from the three dimensional
coordinates only. In this research, we propose an exact prescription to detect
the crystal structures in real-space based on the arrangements of the local
environment and by searching for the directions of possible translational
vectors. The protocol yields good agreements with the experimental results for
any simple or complex crystal structures irrespective of the single-component
or multi-component systems allowing the complete execution to take a minimal
time and computational cost. To the best of our knowledge, this algorithmic
prescription can be applied to detect the unit cell of crystal structure and
identify the space group across the domains of condensed matter physics and
material sciences.

##  I Introduction

Crystal or crystalline solid is the periodic arrangement of constituents
(atoms, molecules, ions etc.) that extends in all directions Kittel (2005).
The detection of crystal structure has been investigated over the last century
in many theoretical or experimental studies and still intriguing in nature
across multiple disciplines of science. X-ray diffraction method was proposed
for more than hundred years ago Friedrich  _et al._ (1912); uk1 (1912); Bragg
(1913); Davisson and Ghrmer (1927), which became the most useful and versatile
technique to detect the crystal structure experimentally Giacovazzo (1992);
Massa (2004). In experiments, the positions and intensities of the Bragg peaks
are measured via X-ray Bragg (1913); Willis (1998); Xie  _et al._ (2013);
Dubach and Guskov (2020); Amann-Winkel  _et al._ (2016), which is one of the
most used approaches to solve the crystal structure currently. Later, the
scattering of neutrons Amann-Winkel  _et al._ (2016) and electrons Deng and
Marks (2006); Saha, Nia, and Rodríguez (2022); Dubach and Guskov (2020) were
also introduced depending on the scientific demands. This data is further fed
to a computer program Computational (1994); Sheldrick (2015); Dolomanov  _et
al._ (2009) to determine the unit cell parameter and space group by analyzing
the properties of the Fourier transform Zanjani  _et al._ (2016); Li, Senesi,
and Lee (2016); Park  _et al._ (2012); Keen and Goodwin (2015); Lu  _et al._
(2012); Harlow ; Siczek  _et al._ (2021); Dawson  _et al._ (2004); Zhu  _et
al._ (2007); Streek and Neumann (2014). The reciprocal lattice arises from the
Fourier transform of another direct or real lattice, which is a periodic
arrangement of atoms or molecules in real space. These formulations are based
on the theory of reciprocal space and it is mostly used in experiments to
handle the detection of crystal structure. The structures of ionic crystals
Son  _et al._ (2004), electronic crystals Monceau (2012), metallic crystals
Hughes and Jain (1979); Tanaka  _et al._ (1999); Pistor and Körner (2021),
nanocrystal superlattices Rogach  _et al._ (2002); Shevchenko  _et al._
(2006); Jones  _et al._ (2010); Zhang  _et al._ (2013); Samanta  _et al._
(2022) etc. were observed in experiments. Very recently, a direct approach of
identifying the ionic colloidal crystal structures was proposed in three
dimensional real-space by matching the index of fluorescently labelled
particles and particle tracking Zang  _et al._ (2024). Multiple experimental
techniques have been developed to detect the crystal structures but
surprisingly in computer simulations, one would face much more difficulties to
identify the crystal structure completely, despite the access of all
coordinates of the system in the real space, where as the crystal is a
periodic arrangement of the coordinates. With the advancement of computer
power, the investigation was started to study various scientific problems
using the computer simulations such as, the phase transition of the solids,
nucleation of the crystals etc. Identification of the crystal structure was
important to solve these crucial problems which turned out to be a regular
scientific exercise in the domain of computational material science or applied
physics. The formation of the crystal structures using different kinds of
idealized models i.e., hard particles Damasceno, Engel, and Glotzer (2012);
Gantapara  _et al._ (2013), patchy particles Zhang  _et al._ (2005); Chen,
Zhang, and Torquato (2018), DNA-mediated nanoparticles O’Brien  _et al._
(2016) etc., were reported in the computer simulations Frenkel and Mulder
(1985); McGrother, Williamson, and Jackson (1996); Bolhuis and Frenkel (1997);
John, Juhlin, and Escobedo (2008); Haji-Akbari  _et al._ (2009). But there was
no straightforward approach to overcome this problem computationally till
date; as a result the problem remains open which demands a much more
attention. The main challenge to detect the crystal structures from computer
simulation is the system size and the equilibration times in the simulation
are orders of magnitude smaller, which is still a huge barrier to identify the
broken symmetry directions in three dimensions. The resolution and quality of
numerical diffraction patterns are too poor to follow the standard protocol
used in the experiments. So, the methods directly depending on the coordinates
of the particles are required to serve the purpose.

A number of different routes based on the real-space analysis, were taken over
the years to analyze the local order including the formation of order
parameter based on the local neighbor Steinhardt, Nelson, and Ronchetti
(1983); Faken and J6nsson (1994); Lechner and Dellago (2008); Damme  _et al._
(2020); Kalian  _et al._ (2023), common-neighbor analysis (CNA) Honeycutt and
Andersen (1987); Faken and J6nsson (1994), templating Duff and Peters (2011),
graph analyses technique Reinhart and Panagiotopoulos (2018); Li  _et al._
(2023). For the last few decades, the crystal structures has been detected
computationally, but no straightforward and general approaches were pointed
out in the literature. The Steinhardt order parameter which was actually
developed to determine the symmetry of local neighbors in dense liquids using
the spherical harmonics, is applicable to a very limited crystal structures
such as Face-Centered Cubic (FCC), Body-Centered Cubic (BCC) or Hexagonal
closed packed (HCP) crystals Steinhardt, Nelson, and Ronchetti (1983).
Recently, a large amount of local fingerprints data was used in machine
learning algorithm Boattini, Dijkstra, and Filion (2019) based on the
detection of topology of the crystallographic environment Spellings and
Glotzer (2018). The authors used the scale-free and rotation-invariant
structural descriptors of a particle’s local environment, which is sensitive
to the symmetry of the local neighborhood. Their observation supports the
detection of structural regions for complex structures but fails to point out
the detail of a complex crystal structure. Artificial intelligence and deep
learning algorithms also enhance the process of the determination of the
crystal structures Ryan, Lengyel, and Shatruk (2018). Recent developments can
distinguish one complex structure from another Spellings and Glotzer (2018);
Boattini  _et al._ (2018), but not capable of producing a particular
crystallographic information without the references. Techniques have been
developed to detect the crystal structures of chemical compounds based on the
chemical spaces Boles, Engel, and Talapin (2016); Bhattacharyya, Makhal, and
Guchhait (2018) or tuning the crystalline behavior in the interaction spaces
Rossi  _et al._ (2011). In computer simulations, despite accessing all the
coordinates of the constituents to measure radial distribution function (RDF),
bond-orientational order (BOO) or diffraction pattern to decipher the
signatures of crystal structures, no general routes were indicated towards the
detection of crystal structures Anders  _et al._ (2015); Zhang  _et al._
(2005); Dshemuchadse, Jung, and Steurer (2011); Bommineni  _et al._ (2019);
Rechtsman, Stillinger, and Torquato (2006); Dshemuchadse  _et al._ (2021);
Dshemuchadse and Steurer (2015); Damasceno, Engel, and Glotzer (2012); Marson
_et al._ (2019); Agarwal and Escobedo (2011); Pan and Dshemuchadse (2023).
Different layers based on bond-orientational order (BOO) were developed to get
more sensitive information of local neighbors but the approaches were
restricted within few structures which did not serve the general purposes
Çamklran, Parsch, and Hibbard (2022); Eslami, Sedaghat, and Müller-Plathe
(2018); Logan  _et al._ (2023). Recently, a theoretical approach was reported
that analyses the point group of the local neighbor directly from the particle
coordinates and shed light on the paths towards the partial detection the
crystal structures Engel (2021). The rotational symmetry of the neighbor was
detected, but it was unable to reveal the complete crystallographic symmetry
and the space group, which remained as open problems. Considering the
shortcomings of the former techniques, automatic identification of the crystal
structure is still unsolved from the real-space analysis only.

In this research, we report a computational protocol to detect the unit cell
and space group of the crystal structure using the three dimensional real-
space analysis. Any other information except the particle coordinates were
ignored to construct the formulation to detect the unit cell of the crystal
structure. The basic difference of the Bravais and non-Bravais lattice is
realized to convert a complex non-Bravais lattice into a Bravais one followed
by the detection of the basis vectors of direct lattice considering the
coordinates of the local neighbors. Based on a few widely used techniques
developed to analyze the crystals, our approach works unambiguously for any
crystal structures with simple or complex basis, beyond the single component
systems and takes minimal time and computational cost. The approach uses
minimum but necessary information in the real space i.e., the coordinates of
the particles only, in order to detect the unit cell and space group which
involves the standard computational “neighbor-search” algorithm in different
ways by minimizing the possibilities step-by-step. Each and every step of the
protocol is presented using an example of ideal crystal structure with complex
basis for the demonstration purpose. The prescription was applied to detect
the unit cells and corresponding space groups of multiple synthetically
prepared crystal structures without noise and the data seemed to agree well
with the experimental results, which are reported here. We also applied the
method on a complex crystal structure coming out of computer simulation and
report the data along with the robustness of the approach at different noise
levels. To the best of our knowledge, this is the first computational approach
of its kind to identify the crystal structures of any ionic crystals, metallic
crystals or covalent network crystals etc. in the real-space which can be
useful in a wider community of condensed matter physics, applied physics or
material sciences.

Figure 1: A schematic diagram of the important steps of protocol is shown.

##  II Methods

The coordinates of all particles in a crystal structure are defined in real-
space. An unit cell is defined as a combination of both the basis vectors and
effective coordinates of the basis that can be replicated to construct the
entire crystal structure by translations only. In three dimensional system, a
crystal structure can be classified into seven crystal classes and fourteen
Bravais lattices within 230 space groups. For a crystal class, the
translational vectors can be different from the primitive lattice vectors
which are constructed by the joining vectors to the adjacent particles. It is
well known that a primitive unit cell is the smallest volume formed by the
primitive lattice vectors which contains only one effective particle inside
the cell. For any primitive Bravais lattice, the basis vectors coincide with
the primitive lattice vectors. A non-primitive Bravais unit cell consists of
more than one effective particles, which are also called “centered unit cell”.
As the choice of the unit cell is not unique for a crystal structure, there
exists multiple possible basis vectors corresponding to the unit cells in
different crystal classes. It follows that one particular crystal class may
include multiple choices of basis vectors depending on the crystal structure
where all the choices are valid for the crystal structure until a particular
choice violates an allowed Bravais unit cell under the corresponding crystal
class. An allowed Bravais unit cell refers to a valid unit cell defined in a
crystal class. For example, a “base-centered” unit cell is not allowed in the
cubic class, hence it is not a rational choice. So, it is important to find
out the crystal class and right choices of basis vectors leading to the
allowed Bravais lattice in that class for the entire crystal.

As we are not interested in identifying the primitive unit cell of the crystal
structure only, we try to evaluate the allowed directions of the basis vectors
for a Bravais unit cell corresponding to the crystal class. If the considered
structure does not satisfy the requirement of Bravais lattice, a
transformation into the Bravais lattice (either primitive or non-primitive) is
required. The key difference between a Bravais and non-Bravais lattice is, all
the particles are equivalent for a Bravais lattice, which is not the case for
a non-Bravais lattice Kittel (2005); Bernstein  _et al._ (2009). In a non-
Bravais lattice, all the particles are non-equivalent, confirming the
existence of multiple types of environment present in the system. This
fundamental difference is exploited to detect a system whether it is Bravais
or non-Bravais in nature. If the considered structure is a non-Bravais one, at
first the system is transformed into a Bravais system and the directions of
basis vectors are identified using the coordinates of the particles in the
Bravais lattice only as discussed in the following sections. Then the unit
cell parameters are detected and the identification of all particles are done
followed by the determination of the space group of the crystal structure. The
primary steps of the methodology are as follows.

  1. 1.

Environment detection of the system and separation of the environment if non-
Bravais lattice

     * •

Evaluation of the radial distribution function (RDF) and the positions of the
local neighbors within a certain distance.

     * •

Determination of the number of clusters formed by the neighbors in presence of
statistical noise and the centroid of the clusters using standard K-Means
clustering method Hartigan and Wong (1979) and separation of the particles
with similar environment.

     * •

Bravais lattice if all particles have similar kind of environment, otherwise
non-Bravais lattice. If non-Bravais lattice, transform the system into a
Bravais lattice.

  2. 2.

Considering the Bravais lattice as the system, detection of crystal class :
all possible directions of vectors

     * •

Construction of a convex polyhedron using the centroid of the clusters of the
distribution of local neighbors.

  3. 3.

Evaluation of the set consisting of the directions of all possible basis
vectors for the Bravais lattice under the crystal class

  4. 4.

Construction of the unit cell of considered crystal structure for each choice
of basis vectors satisfying Bravais unit cell

  5. 5.

Determination of the space group of the crystal structure

A schematic diagram of the approach is shown in Fig. 1 indicating all the
primary and necessary steps starting from the consideration of the particle
coordinates of the system up to the determination of the unit cell and space
group of a crystal structure. The steps are elaborated to accomplish the
detection of the crystal structure using the ideas plugged into the course. To
demonstrate the mechanism of the approach, a synthetically prepared ideal
system of $\beta$-Mn crystal was used with lattice parameters a=6.315,
b=6.315, c=6.315, $\alpha$=90∘, $\beta$=90∘, $\gamma$=90∘ and space group
$P4_{1}32$, which was verified experimentally Xie  _et al._ (2013). The unit
cell of the $\beta$-Mn crystal consisting of twenty particles (without any
noise) was chosen to replicate in the three dimensions, resulting an ideal
$\beta$-Mn crystalline system, as shown in Fig. 2A. We used only the
coordinates of the particles and any other information was lost intentionally.
Our final goal was to get one of the right choices of basis vectors and unit
cell back for the crystal structure.

Figure 2: (A) An ideal system of $\beta$-Mn crystal is shown. (B) The
distributions of neighbors within the distance $r_{c}$ $\sim$ 2.6 were
considered for three arbitrarily chosen particles from the crystal structure,
showing completely different kind of distributions for the particles.

###  II.1 Detection and separation of the environment : Bravais or non-Bravais
lattice and conversion of a non-Bravais lattice into a Bravais lattice

As the first step of the scheme, a crystal structure is checked whether it is
a Bravais lattice or not. The configuration shown in Fig. 2A is chosen as the
system of the interest and the position of all particles are accessed in
three-dimensional Cartesian coordinate system. We detect types of the
positional environment for all particles in the system and separate the
particles sharing similar environment following a technique namely
“environment separation” which is as described below.

In principle, the environment of a particle can be realized by the positional
distribution of the neighbor particles around itself. A simple protocol
includes choosing a particle from the system randomly followed by the
observation of the distribution of the neighbor particles around itself within
a certain distance. Subsequently, multiple kinds of environment correspond to
the existence of different kinds of positional distribution of the neighbors
around the particles in the system estimated within the fixed distance. So, a
simple realization is that if there exists more than one kind of positional
arrangement of the neighbors for all the particles in the system within a
fixed distance, the system is said to be a non-Bravais lattice. If all the
particles have similar kind of distribution of the neighbors, the system is
Bravais in nature as all the particles in the system are equivalent. For the
instance, a system of ideal $\beta$-Mn crystal structure is showcased in Fig.
2A. The positional distributions of the neighbors within a certain distance
($\sim$ 2.6) for all particles in the system were obtained upon performing a
simple “neighbor search” calculation using freud-toolkit Ramasubramani  _et
al._ (2020). We observed multiple kind of distributions of the neighbors and
the distributions for only three arbitrarily chosen particles from the system
are shown in Figs. 2B, C, D for the demonstration purpose. Only visual
inspection confirms that the three randomly selected particles have different
kinds of environment without further quantitative data. An obvious statement
can be made that more than one kind of environment exists in the system of
$\beta$-Mn crystal; so the system is a non-Bravais lattice. As we understand
that only visual inspection is not enough to detect the types of the
environment if the system is more complex or noisy; we need to use a general
and robust algorithmic approach to detect the environment. As the output of
this scrutiny, a system is said to be either Bravais or non-Bravais lattice.

Figure 3: (A) An ideal system of $\beta$-Mn crystal (complex basis) and the
corresponding RDF (B) is shown. Detecting the first minimum, i.e. nearest
neighbor distance $\sim$ 2.6, (C) the position of the atoms within $r_{c}$ has
been evaluated confirming eight unique clouds of particles from K-Means
clustering as shown in (D).

For a non-Bravais lattice, the “environment separation” technique is
decomposed into two sub-categories, (i) positional distribution of the local
neighbors of all particles in the system and (ii) separation of the particles
based on the unique environments after clustering the points of the three
dimensional distribution of the neighbors. These steps are explained below in
detail.

####  II.1.1 Positional distribution of the local neighbors

As a part of the “environment detection” technique, at first it is required to
know the positional distribution of the local neighbors of all particles in
the system. We consider a system with $N$ particles. The radial distribution
function (RDF) is evaluated considering the geometric centers of particles
measuring the normalized particle density as a function of distance from a
central particle as shown in Fig. 3B using the freud-toolkit Ramasubramani
_et al._ (2020). $N_{CN}$ is the number of neighbors for each particle within
the distance, $r_{c}$, which is chosen to be a minima of RDF. In general the
first minima of RDF is preferred to calculate the positional distributions of
the nearest neighbors in three dimensions. For this example, $r_{c}$ was
chosen as 2.6 and $N_{CN}$ appeared to be 13. All the pairwise position
vectors of the particles are calculated within the distance $r_{c}$, using
$\vec{r}_{ij}=\vec{r}_{j}-\vec{r}_{i}$, where $i$ $\in$ ($0,N$) and $j$ $\in$
($0,N_{CN}$) for particle i, followed by the superposition of all vectors
$\vec{r}_{ij}$. This gives the overall positional arrangement of all the
neighbors for all the particles in the system, which can be considered as the
bond orientational order (BOO) of the system in three dimensions, as shown in
Fig. 3C. For an ideal crystal, a distribution of the particles with zero
positional deviation is obtained but in real simulation data, few cluster of
particles are observed with statistical noise, located at the distance $\sim$
$r_{c}$, due to the translational symmetry of the crystal structure.

####  II.1.2 Clustering and environment detection

The number of clusters ($N_{c}$) is calculated using the K-Means clustering
method Hartigan and Wong (1979) by detecting the elbow of the two-dimensional
plot of the sum of the square distance between particles in a cluster and the
cluster centroid (WCSS stands for “Within-Cluster Sum of Square”) versus
k-value (x-axis) as the standard approach, as shown in Fig. 3D. The elbow is
determined as a number in the x-axis after which, the value of WCSS decreases
almost monotonically. The centroid of $N_{c}$ clusters are evaluated with the
positions around the centre (0, 0, 0). For an ideal system, the centroid of
the clusters are not required to calculate due to the absence of any
positional fluctuations as in the real system leading to the coordinate of the
centroid of a cluster coinciding with the points. But in real simulation data,
the identification of the clusters is important followed by the determination
of the centroid of the clusters. For ideal $\beta$-Mn structure, $N_{c}$
($\neq$ $N_{CN}$ = 13) was 48 as shown in the elbow analysis (Fig. 3D).
Additionally, the cluster ids of all $\vec{r}_{ij}$ vectors in the bond order
diagram are identified to know the environment of all the particles in the
system.

The environment of a particle i is characterized by the cluster ids of
neighbors around itself within a distance $r_{c}$. $\zeta$ corresponds to the
environments of all particles in the system and $\zeta_{im}$ indicates the
cluster id of $m$-th neighbor of $i$-th particle, where $i$ $\in$ ($0,N$) and
$m$ $\in$ ($0,N_{CN}$) within $r_{c}$ distance. So $\zeta_{i}$ refers to the
set of cluster ids of all $N_{CN}$ neighbors of $i$-th particle. As multiple
particles can possess similar environment in a crystal structure, we find the
particles with similar type of environment i.e., unique $\zeta_{i}$ based on
the integer values (cluster ids of all neighbors) by varying $i$ from $0$ to
$N$ consecutively. For example, if $\zeta_{1}$ and $\zeta_{2}$ defining the
cluster ids of the neighbors of particle 1 and 2 respectively, share identical
values then these two particles are said to have similar environment in an
ideal system. From the system, unique $\zeta_{i}$s are sorted out and total
number of unique $\zeta_{i}$ is a quantification of the types of environment
for all particles in the system. The particle ids having identical $\zeta_{i}$
(any one among the unique $\zeta_{i}$s) are identified and total number of
such particles sharing similar environment is denoted by $N_{s}$. If the
initial system is an ideal Bravais lattice then all $\zeta_{i}$ are identical
i.e., $N_{s}=N$, as all particles in a Bravais lattice share similar kind of
environment and those are equivalent. In the simulated system, $N_{s}$ becomes
almost comparable with $N$; $N_{s}\sim N$, due to the noise present in the
system. In a non-Bravais lattice, as there exists multiple types of
environment, $N_{s}$ is very less than $N$ confirming the existence of more
than one unique $\zeta_{i}$ for all the particles. The formulation indicates
that $N_{CN}\leq N_{c}$ where $N_{CN}$ is the number of neighbors of each
particle in the system within $r_{c}$ and $N_{c}$ corresponds to the total
number of clusters formed by all particles in the system. For a non-Bravais
lattice, more than one kind of environment suggests the presence of extra
clusters than that of a Bravais lattice, as all particles in the Bravais
system share similar environment following the number of clusters $N_{c}$ of
all particles to be equal with $N_{CN}$. Following the recipe, the decision
can be made by comparing the number $N_{s}$ with $N$ barring the noise; if
$N_{s}\sim N$, the system is called a Bravais system, otherwise it is a non-
Bravais one. For the ideal $\beta$-Mn system shown in Fig. 2A, total number of
environments appeared to be twenty for all particles in the system within the
distance $r_{c}$ = 2.6. $N_{s}$ was found to be very less than $N$ ($N_{s}$ =
0.05 $\times$ $N$) indicating the existence of multiple types of environment
confirming the system as a non-Bravais lattice. Figs. 2B, C, D also suggest
the existence of different kinds of environment for three arbitrarily chosen
particles from the ideal $\beta$-Mn system, where the local neighbors of the
particles within the distance $r_{c}$ were shown separately in each sub-
figure.

Figure 4: The schematic diagram of the “environment separation” technique is
shown

####  II.1.3 Environment separation of a non-Bravais lattice

If the initial structure appears to be a non-Bravais lattice, it is compulsory
to perform the “environment separation” technique to transform the non-Bravais
system into a Bravais one, which is not an essential requirement if the
initial system is a Bravais lattice. To perform the “environment separation”,
we need to consider an updated system with $N_{s}$ particle coordinates only,
which are obtained upon separating the $N$ particles in the initial system
based on the environments. This does not necessarily mean that all the $N_{s}$
particles in the transformed system will have similar kind of environment
around themselves, when only $N_{s}$ particles are considered as the system.
For the confirmation check the coordinates of the $N_{s}$ particles are
considered as the transformed system (total number of particle in the updated
system is $N_{s}$ i.e., $N=N_{s}$) and all the previous steps consisting of
the calculation of RDF, BOO, K-Means clustering are applied sequentially
again, followed by the categorization of particles based on the updated
environments and separation of the $N_{s}$ particles with similar kind of
environment. If $N_{s}=N$, the system is transformed into a Bravais lattice,
but if not then the protocol needs to be performed again. So, this process is
performed in an iterative way and at each step of the iteration, the
coordinates of $N_{s}$ particles coming out of the previous environment
separation are considered as the transformed system until all the particles in
the transformed system have similar kind of environment. The iteration for the
“environment separation” process discontinues if the number of particles
having similar kind of environment matches with the number of particles
considered as the system at the start of that particular step of iteration
barring the noise. The transformed Bravais lattice does not necessarily
required to be a primitive cell. The schematic diagram of the “environment
separation” technique is shown in Fig. 4 and the number of iteration for the
separation protocol depends on the number of environments present in the
initial system within the distance $r_{c}$. For a non-Bravais lattice, the
system changes into a Bravais lattice after performing all the required steps;
while Bravais lattice remain unchanged, which eventually does not require any
conversion. In principle, a Bravais lattice can always be extracted from a
non-Bravais lattice irrespective of the existence of multiple kinds of
environment present in the system. The algorithmic representation of the
“environment separation” technique is described below.

$N\text{ particle coordinates}$

RDF

BOO within r_c

K-Means clsutering with local neighbors

$\text{Calculate }\zeta\text{ for all particles}$

$\text{Separate }N_{s}\text{ particles based on }\zeta_{i}\text{, where
}i\rightarrow 0\text{, }N$

while $N_{s}$ $<$ $N$ do

$N\leftarrow N_{s}$ (Update $N$)

$\text{Updated system with }N\text{ particles}$

RDF

BOO within r_c

K-Means clsutering with local neighbors

$\text{Calculate }\zeta\text{ for all particles in the updated system}$

$\text{Separate }N_{s}\text{ particles based on }\zeta_{i}\text{, where
}i\rightarrow 0\text{, }N$

end while

Figure 5: (A) An ideal system of $\beta$-Mn crystal, (B) the same system with
atoms in multiple colors based on the similar kind of environment and (C) a
reduced Bravais lattice after environment separation are shown.

After the detection of the environment, the “environment separation" method is
performed in the iterative way if required; a non-Bravais lattice converts
into a Bravais lattice, as the complex $\beta$-Mn crystal transforms into a
“simple cubic" lattice within a single iteration as shown in Fig. 5. For this
example, the transformation of a non-Bravais lattice into a primitive Bravais
lattice does not necessarily mean that the transformation will always occur in
primitive Bravais lattice like the $\beta$-Mn structure; in principle, the
transformation may convert a non-Bravais lattice into any of the fourteen
Bravais lattices. The $\beta$-Mn crystal structure is shown with the particles
in single color (Fig. 5A), followed by the categorization of the particles
based on the environments as shown in Fig. 5B, indicating the particles with
similar environment in the same color. The particles with any particular color
(for example, the particles in “blue” color) are chosen and considered as the
system for further analysis. The choice of the particular set of particles
does not depend on the type of the Bravais lattice; any particular choice
satisfies the same Bravais lattice. It is important to note, the chosen
coordinates of the particles with similar environment are already present in
the initially considered non-Bravais $\beta$-Mn crystal structure and our goal
is to find those particles following the protocol described above.

###  II.2 Identification of crystal class and all possible directions

In this subsection, our aim is to get the directions of all possible vectors,
$\hat{\mathbf{a}}_{d}$, $\hat{\mathbf{b}}_{d}$, $\hat{\mathbf{c}}_{d}$, which
can be considered as one of the possible choices of the basis vectors. In
general, all the possible choices may not be valid as some of those fail to
produce the allowed Bravais lattices among fourteen, defined under one of the
seven crystal classes. The transformed Bravais lattice (here, “simple cubic”
lattice for $\beta$-Mn structure) is considered as the system for the time
being and used in next few steps. This approach to identify the crystal class
is decomposed into further sub-categories; the construction of a convex
polyhedron using the centroid of the clusters in the BOO after the K-Means
clustering and evaluation of all possible directions from the polyhedron.

####  II.2.1 Construction of convex polyhedron from the positional
distribution of all neighbors

The transformed Bravais lattice is considered as the reduced structure and the
system of interest, which is used to evaluate the positional distribution of
the local neighbors. We perform exactly similar analyses including the
determination of RDF (Fig. 6A) with a further selection of the first minimum
of the RDF, $r_{c}$ $\sim$ 6.5. The pairwise position vectors $\vec{r}_{ij}$
are calculated using the coordinates of the particles producing a completely
different type of positional distribution for the “simple cubic” Bravais
lattice as shown in Fig. 6B.

After applying the K-Means clustering method on the positional distribution,
the number of clusters turns out to be $N^{\prime}_{c}$ (= 6 for this example
as shown in Fig. 6C). The centroid of the $N^{\prime}_{c}$ clusters is further
used to construct a convex polyhedron. A similar approach was reported earlier
in the context of crystal structure detection by matching the templates of the
local neighbors Larsen, Schmidt, and SchiØtz (2016). In this example, the
constructed polyhedron is an Octahedron with six vertices as shown in Fig. 6D.
The constructed polyhedron may not be regular always, but must have the
property of convexity. For the simulated system with noise, the constructed
polyhedron can not be a regular one, but the convexity always remains there
within the tolerances.

Figure 6: (A) The RDF, (B) BOO are shown for the Bravais lattice. The number
of clusters is six as confirmed by the elbow analysis of K-Means clustering
(C), and (D) constructed convex polyhedron using the centroid of the clusters
are shown. Figure 7: Synthetically prepared crystal structures of the Bravais
lattices corresponding to (A) Cubic, (B) Hexagonal and (C) Tetragonal classes
are shown. The corresponding unit cells of the structures and the convex
polyhedra constructed from the distribution of the local neighbors along with
the directions of the lattice vectors, $\hat{a}$, $\hat{b}$, $\hat{c}$ are
displayed for each case. The letters P, I, B, F designate the primitive, body-
centered, any base-centered and all face-centered Bravais unit cell
respectively for each crystal class. Figure 8: Synthetically prepared crystal
structures of the Bravais lattices corresponding to (A) Trigonal, (B)
Orthorhombic and (C) Monoclinic and (D) Triclinic crystal classes are shown.
The corresponding unit cells of the structures and the convex polyhedra
constructed from the distribution of the local neighbors along with the
directions of the lattice vectors, $\hat{a}$, $\hat{b}$, $\hat{c}$ are
displayed for each case. The letters P, I, B, F designate the primitive, body-
centered, any base-centered and all face-centered Bravais unit cell
respectively for each crystal class.

####  II.2.2 Evaluation of all possible directions from the polyhedron

For any of the fourteen Bravais lattices, the directions of the three lattice
vectors i.e. $\hat{\mathbf{a}}_{d}$, $\hat{\mathbf{b}}_{d}$,
$\hat{\mathbf{c}}_{d}$ appear to intersect either any vertices, or face mid-
points or edge mid-points of the polyhedron constructed with the centroid of
the neighbors of the Bravais lattice as shown in Figs. 7, 8. The polyhedron
may change depending on the distance $r_{c}$ considered to calculate the
neighbors, but the directions of the lattice vectors always intersect either
any of the vertices or face-midpoints or edge-midpoints. It follows that if
the Bravais lattice is a primitive unit cell with only one effective particle
per unit cell, the directions of the basis vectors intersect any of the
vertices of the constructed polyhedron. In principle, the directions of
rotational symmetry axes of the polyhedron point group can be considered for
the structures with crystal class of higher symmetries but this consideration
is not effective for the crystal classes with lower number of rotational
symmetry like Monoclinic or Triclinic. In order to calculate the face mid-
points and edge mid-points of the polyhedron, the convex-decomposition
technique of Coxeter toolkit is used Ramasubramani  _et al._ (2021). The
vertices, forming each face and edge are determined followed by the
calculation of the geometric centres of the faces and edge mid-points. The
coordinates of all the vertices, face-midpoints and edge-midpoints are known
around the center of the constructed polyhedron and all the joining vectors
are non-unit vectors. $g$ is the total number of joining vectors passing
through any of these attributes of polyhedron with origin at (0, 0, 0) and all
the vectors are considered for the determination of the direction of lattice
vectors.

Each triplet of the ${}^{g}C_{3}$ combinations is considered as a set of
$\vec{\mathbf{a}}_{d}$, $\vec{\mathbf{b}}_{d}$, $\vec{\mathbf{c}}_{d}$ and
used to detect the crystal class among seven, by measuring the
$|\vec{\mathbf{a}}_{d}|$, $|\vec{\mathbf{b}}_{d}|$, $|\vec{\mathbf{c}}_{d}|$
and the mutual angles defined as $\alpha$, $\beta$ and $\gamma$ within a
distance ($\mathcal{X}_{d}$) and angle ($\mathcal{X}_{a}$) tolerance for the
real system (for an ideal system, $\mathcal{X}_{d}$=0 and
$\mathcal{X}_{a}$=0). The seven crystal classes are considered according to
the order of the crystallographic point group symmetry; cubic (48), hexagonal
(24), tetragonal (16), trigonal (12), orthorhombic (8), monoclinic (4) and
triclinic (2). Each triplet consisting of three non-unit vectors, is checked
computationally to satisfy the criteria of any of the crystal classes. The
necessary conditions of satisfying the seven crystal systems based on the
lattice parameters $|\vec{\mathbf{a}}_{d}|$, $|\vec{\mathbf{b}}_{d}|$,
$|\vec{\mathbf{c}}_{d}|$, $\alpha$, $\beta$, $\gamma$ are discussed below.

  1. 1.

Cubic $\rightarrow$ $||\vec{a}_{d}|-|\vec{b}_{d}||\leq\mathcal{X}_{d}$,
$||\vec{a}_{d}|-|\vec{c}_{d}||\leq\mathcal{X}_{d}$,
$||\vec{b}_{d}|-|\vec{c}_{d}||\leq\mathcal{X}_{d}$,
$|\alpha-\beta|\leq\mathcal{X}_{a}$, $|\gamma-\beta|\leq\mathcal{X}_{a}$,
$|\alpha-\gamma|\leq\mathcal{X}_{a}$,
$(90^{\circ}-\mathcal{X}_{a})\leq\alpha\leq(90^{\circ}+\mathcal{X}_{a})$,
$(90^{\circ}-\mathcal{X}_{a})\leq\beta\leq(90^{\circ}+\mathcal{X}_{a})$,
$(90^{\circ}-\mathcal{X}_{a})\leq\gamma\leq(90^{\circ}+\mathcal{X}_{a})$

  2. 2.

Hexagonal $\rightarrow$ (i)
$||\vec{a}_{d}|-|\vec{b}_{d}||\leq\mathcal{X}_{d}$,
$||\vec{a}_{d}|-|\vec{c}_{d}||>\mathcal{X}_{d}$,
$||\vec{b}_{d}|-|\vec{c}_{d}||>\mathcal{X}_{d}$,
$(90^{\circ}-\mathcal{X}_{a})\leq\alpha\leq(90^{\circ}+\mathcal{X}_{a})$,
$(90^{\circ}-\mathcal{X}_{a})\leq\beta\leq(90^{\circ}+\mathcal{X}_{a})$,
$(120^{\circ}-\mathcal{X}_{a})\leq\gamma\leq(120^{\circ}+\mathcal{X}_{a})$

(ii) $||\vec{a}_{d}|-|\vec{c}_{d}||\leq\mathcal{X}_{d}$,
$||\vec{a}_{d}|-|\vec{b}_{d}||>\mathcal{X}_{d}$,
$||\vec{b}_{d}|-|\vec{c}_{d}||>\mathcal{X}_{d}$,
$(90^{\circ}-\mathcal{X}_{a})\leq\alpha\leq(90^{\circ}+\mathcal{X}_{a})$,
$(120^{\circ}-\mathcal{X}_{a})\leq\beta\leq(120^{\circ}+\mathcal{X}_{a})$,
$(90^{\circ}-\mathcal{X}_{a})\leq\gamma\leq(90^{\circ}+\mathcal{X}_{a})$

(iii) $||\vec{b}_{d}|-|\vec{c}_{d}||\leq\mathcal{X}_{d}$,
$||\vec{a}_{d}|-|\vec{b}_{d}||>\mathcal{X}_{d}$,
$||\vec{a}_{d}|-|\vec{c}_{d}||>\mathcal{X}_{d}$,
$(120^{\circ}-\mathcal{X}_{a})\leq\alpha\leq(120^{\circ}+\mathcal{X}_{a})$,
$(90^{\circ}-\mathcal{X}_{a})\leq\beta\leq(90^{\circ}+\mathcal{X}_{a})$,
$(90^{\circ}-\mathcal{X}_{a})\leq\gamma\leq(90^{\circ}+\mathcal{X}_{a})$

  3. 3.

Tetragonal $\rightarrow$ $\rightarrow$ (i)
$||\vec{a}_{d}|-|\vec{b}_{d}||\leq\mathcal{X}_{d}$,
$||\vec{a}_{d}|-|\vec{c}_{d}||>\mathcal{X}_{d}$,
$||\vec{b}_{d}|-|\vec{c}_{d}||>\mathcal{X}_{d}$,
$|\alpha-\beta|\leq\mathcal{X}_{a}$, $|\gamma-\beta|\leq\mathcal{X}_{a}$,
$|\alpha-\gamma|\leq\mathcal{X}_{a}$,
$(90^{\circ}-\mathcal{X}_{a})\leq\alpha\leq(90^{\circ}+\mathcal{X}_{a})$,
$(90^{\circ}-\mathcal{X}_{a})\leq\beta\leq(90^{\circ}+\mathcal{X}_{a})$,
$(90^{\circ}-\mathcal{X}_{a})\leq\gamma\leq(90^{\circ}+\mathcal{X}_{a})$

(ii) $||\vec{a}_{d}|-|\vec{c}_{d}||\leq\mathcal{X}_{d}$,
$||\vec{a}_{d}|-|\vec{b}_{d}||>\mathcal{X}_{d}$,
$||\vec{b}_{d}|-|\vec{c}_{d}||>\mathcal{X}_{d}$,
$|\alpha-\beta|\leq\mathcal{X}_{a}$, $|\gamma-\beta|\leq\mathcal{X}_{a}$,
$|\alpha-\gamma|\leq\mathcal{X}_{a}$,
$(90^{\circ}-\mathcal{X}_{a})\leq\alpha\leq(90^{\circ}+\mathcal{X}_{a})$,
$(90^{\circ}-\mathcal{X}_{a})\leq\beta\leq(90^{\circ}+\mathcal{X}_{a})$,
$(90^{\circ}-\mathcal{X}_{a})\leq\gamma\leq(90^{\circ}+\mathcal{X}_{a})$

(iii) $||\vec{b}_{d}|-|\vec{c}_{d}||\leq\mathcal{X}_{d}$,
$||\vec{a}_{d}|-|\vec{b}_{d}||>\mathcal{X}_{d}$,
$||\vec{a}_{d}|-|\vec{c}_{d}||>\mathcal{X}_{d}$,
$|\alpha-\beta|\leq\mathcal{X}_{a}$, $|\gamma-\beta|\leq\mathcal{X}_{a}$,
$|\alpha-\gamma|\leq\mathcal{X}_{a}$,
$(90^{\circ}-\mathcal{X}_{a})\leq\alpha\leq(90^{\circ}+\mathcal{X}_{a})$,
$(90^{\circ}-\mathcal{X}_{a})\leq\beta\leq(90^{\circ}+\mathcal{X}_{a})$,
$(90^{\circ}-\mathcal{X}_{a})\leq\gamma\leq(90^{\circ}+\mathcal{X}_{a})$

  4. 4.

Trigonal $\rightarrow$ $||\vec{a}_{d}|-|\vec{b}_{d}||\leq\mathcal{X}_{d}$,
$||\vec{a}_{d}|-|\vec{c}_{d}||\leq\mathcal{X}_{d}$,
$||\vec{b}_{d}|-|\vec{c}_{d}||\leq\mathcal{X}_{d}$,
$|\alpha-\beta|\leq\mathcal{X}_{a}$, $|\gamma-\beta|\leq\mathcal{X}_{a}$,
$|\alpha-\gamma|\leq\mathcal{X}_{a}$, $\alpha\leq(90^{\circ}-\mathcal{X}_{a})$
or $\alpha>(90^{\circ}+\mathcal{X}_{a})$,
$\beta\leq(90^{\circ}-\mathcal{X}_{a})$ or
$\beta>(90^{\circ}+\mathcal{X}_{a})$, $\gamma\leq(90^{\circ}-\mathcal{X}_{a})$
or $\gamma>(90^{\circ}+\mathcal{X}_{a})$

  5. 5.

Orthorhombic $\rightarrow$ $||\vec{a}_{d}|-|\vec{b}_{d}||>\mathcal{X}_{d}$,
$||\vec{a}_{d}|-|\vec{c}_{d}||>\mathcal{X}_{d}$,
$||\vec{b}_{d}|-|\vec{c}_{d}||>\mathcal{X}_{d}$, $\rightarrow$
$|\alpha-\beta|\leq\mathcal{X}_{a}$, $|\gamma-\beta|\leq\mathcal{X}_{a}$,
$|\alpha-\gamma|\leq\mathcal{X}_{a}$,
$(90^{\circ}-\mathcal{X}_{a})\leq\alpha\leq(90^{\circ}+\mathcal{X}_{a})$,
$(90^{\circ}-\mathcal{X}_{a})\leq\beta\leq(90^{\circ}+\mathcal{X}_{a})$,
$(90^{\circ}-\mathcal{X}_{a})\leq\gamma\leq(90^{\circ}+\mathcal{X}_{a})$

  6. 6.

Monoclinic $\rightarrow$ (i) $||\vec{a}_{d}|-|\vec{b}_{d}||>\mathcal{X}_{d}$,
$||\vec{a}_{d}|-|\vec{c}_{d}||>\mathcal{X}_{d}$,
$||\vec{b}_{d}|-|\vec{c}_{d}||>\mathcal{X}_{d}$,
$|\alpha-\beta|\leq\mathcal{X}_{a}$, $|\gamma-\beta|\geq\mathcal{X}_{a}$,
$|\alpha-\gamma|\geq\mathcal{X}_{a}$,
$(90^{\circ}-\mathcal{X}_{a})\leq\alpha\leq(90^{\circ}+\mathcal{X}_{a})$,
$(90^{\circ}-\mathcal{X}_{a})\leq\beta\leq(90^{\circ}+\mathcal{X}_{a})$,
$(90^{\circ}+\mathcal{X}_{a})\leq\gamma$

(ii) $||\vec{a}_{d}|-|\vec{b}_{d}||>\mathcal{X}_{d}$,
$||\vec{a}_{d}|-|\vec{c}_{d}||>\mathcal{X}_{d}$,
$||\vec{b}_{d}|-|\vec{c}_{d}||>\mathcal{X}_{d}$,
$|\alpha-\gamma|\leq\mathcal{X}_{a}$, $|\gamma-\beta|\geq\mathcal{X}_{a}$,
$|\alpha-\beta|\geq\mathcal{X}_{a}$,
$(90^{\circ}-\mathcal{X}_{a})\leq\alpha\leq(90^{\circ}+\mathcal{X}_{a})$,
$(90^{\circ}+\mathcal{X}_{a})\leq\beta$,
$(90^{\circ}-\mathcal{X}_{a})\leq\gamma\leq(90^{\circ}+\mathcal{X}_{a})$

(iii) $||\vec{a}_{d}|-|\vec{b}_{d}||>\mathcal{X}_{d}$,
$||\vec{a}_{d}|-|\vec{c}_{d}||>\mathcal{X}_{d}$,
$||\vec{b}_{d}|-|\vec{c}_{d}||>\mathcal{X}_{d}$,
$|\beta-\gamma|\leq\mathcal{X}_{a}$, $|\gamma-\alpha|\geq\mathcal{X}_{a}$,
$|\alpha-\beta|\geq\mathcal{X}_{a}$, $(90^{\circ}+\mathcal{X}_{a})\leq\alpha$,
$(90^{\circ}-\mathcal{X}_{a})\leq\beta\leq(90^{\circ}+\mathcal{X}_{a})$,
$(90^{\circ}-\mathcal{X}_{a})\leq\gamma\leq(90^{\circ}+\mathcal{X}_{a})$

  7. 7.

Triclinic $\rightarrow$ $||\vec{a}_{d}|-|\vec{b}_{d}||>\mathcal{X}_{d}$,
$||\vec{a}_{d}|-|\vec{c}_{d}||>\mathcal{X}_{d}$,
$||\vec{b}_{d}|-|\vec{c}_{d}||>\mathcal{X}_{d}$,
$|\alpha-\beta|>\mathcal{X}_{a}$, $|\alpha-\gamma|>\mathcal{X}_{a}$,
$|\beta-\gamma|>\mathcal{X}_{a}$

Figure 9: (A) The direction of the lattice vectors $\hat{a}$, $\hat{b}$ and
$\hat{c}$, passing through the vertices of the constructed Octahedron are
shown. The coordinate of the neighbors without the Octahedron along with
$\hat{a}$, $\hat{b}$ and $\hat{c}$ as shown in (B), indicates the direction of
lattice vectors as the valid choice of unit cell for the Simple Cubic Bravais
lattice, displayed in (C).  Figure 10: (A) The non-Bravais lattice system with
$N$ particle coordinates are shown. (B) A random chunk of particles from the
system are chosen and the direction of the lattice vectors, $\hat{a}$,
$\hat{b}$ and $\hat{c}$ are shown along with the unit cell for the validation
purpose.

According to the observations, the crystal class of the Bravais lattice always
involves the highest order crystallographic point group satisfied by the
triplets from ${}^{g}C_{3}$ combinations within the distance tolerance
$\mathcal{X}_{d}$ and angle tolerance $\mathcal{X}_{a}$. For example, if a
system consists of triplets satisfying the cubic, tetragonal or monoclinic
crystal classes, only those triplets corresponding to the cubic class are
considered, as the cubic is of the highest order crystal class present in the
system. All other triplets obeying lower symmetry crystal classes or no
classes are ignored. We sort out those triplets corresponding to the highest
order crystal class only which are considered as a set $\mathcal{T}$ and used
for further analysis. Multiple triplets corresponding to the same crystal
class of the highest order can be present in the system depending on the
crystal class. Each triplet of the set $\mathcal{T}$ is capable of producing
the entire Bravais lattice by translating a particle along each of the three
directions. For the validation purpose in the ideal $\beta$-Mn system, we
observed only one choice of the directions of the vectors, which was chosen as
the directions of basis vectors by default. Those directions are shown in Fig.
9A with the constructed Octahedron using the particles in the BOO. The same
particle coordinates of the neighbors in BOO are presented in Fig. 9B without
the constructed polyhedron indicating the directions of the basis vectors to
replicate the “simple cubic” (SC) Bravais lattice by translational only upon
associating a random particle with a lattice site (Fig. 9C). This particular
choice can be regarded as primitive lattice vectors because “simple cubic” is
a valid Bravais unit cell in the cubic class, which is also a primitive cell.
The particles shown in the deep “blue” color in Fig. 9C also supports the
choice of the basis vectors. This choice is also valid even for the initial
non-Bravais crystal structure as shown in Fig. 10.

The choices of the directions of the basis vectors do not depend whether these
directions commensurate with the simulation box vectors or not, as the
identification involves the local coordinate of the particles in the system.
Though for this example, it is the case, but for a simulated system, this does
not happen generally. The discussed methodology works well irrespective of
this fact and does not get affected in the simulation data as well. One such
example is discussed below as in the “Results” section.

###  II.3 Identification of the directions of basis vectors of Bravais lattice

We consider the Bravais lattice and the triplets of set $\mathcal{T}$ leading
to the construction of the entire lattice. All the directions of triplets may
not be the rational choices for the allowed Bravais unit cells (primitive or
non-primitive) under the detected crystal class. So the directions of triplets
satisfying the conditions are sorted out and those are the directions of basis
vectors ($\hat{\mathbf{a}}$, $\hat{\mathbf{b}}$, $\hat{\mathbf{c}}$). The
detail protocol of separating all the directions of basis vectors from the set
$\mathcal{T}$ are discussed below.

####  II.3.1 Coordinates of the corner particles of the parallelepiped

A triplet is chosen from the set $\mathcal{T}$ and the directions of the
triplet vectors are estimated followed by the association of a random particle
from the Bravais lattice to a lattice site. The other corner particles of the
parallelepiped are searched along each of the three directions of the
considered vectors one by one in order to get the nearest particles.
Association of a particle to a lattice site does not affect the space group of
the crystal structure upon successful determination of the unit cell. This
process helps us to get all the particles, located at the corners of the
parallelepiped by searching the nearest one along a direction of the three
lattice vectors as shown in cartoon figure (Fig. 11A).

Starting from a randomly chosen particle, say $N_{0}$, we get total four
particles located at the corners of the parallelepiped, $N_{1}$, $N_{2}$,
$N_{3}$ at the minimum distances, joined via $\hat{a}$, $\hat{b}$, $\hat{c}$
respectively. $N_{4}$ can be obtained by translating along $\hat{c}$ from
$N_{1}$. The same protocol suggests, $N_{5}$ is connected with $N_{3}$ via
$\hat{b}$ and $N_{6}$ is attached with $N_{1}$ via $\hat{b}$ at the minimum
distances respectively as illustrated in Fig. 11A. The same figure also
confirms that $N_{7}$ is directly connected to $N_{6}$ via $\hat{c}$. In this
way, the identities and coordinates of these eight corner particles of the
parallelepiped are obtained. But the identities are not needed to be followed
strictly as described here but one should stick to any particular
representation for the rest of the approach. It should be noted that we use
the transformed Bravais lattice system in case the initial system is non-
Bravais in nature. Here, we identify the parallelepiped with eight corner
particles, not the conventional Bravais lattice, which may or may not be a
primitive cell. The algorithmic way to detect the eight corners of the
parallelepiped is the following.

$\hat{a},\hat{b},\hat{c}\text{ and coordinates of all particles in Bravais
lattice}$

$\mathcal{P}=[]$

$\mathcal{L}=[\hat{a},\hat{b},\hat{c}]$

Choose a random particle, c

for $l$ in $\mathcal{L}$ do

$\mathcal{Q}=[]$

for $i$ $\rightarrow$ (0, N) do

$\vec{v}$ = $\vec{r}_{i}-\vec{r}_{c}$

$\hat{v}=\vec{v}/|\vec{v}|$

$\theta=\cos^{-1}(\hat{v}\cdot l)$

if $\theta\leq\mathcal{X}_{a}$ then

$\mathcal{Q}\leftarrow i$

end if

end for

$\text{dist }=[]$

for $j$ in $\mathcal{Q}$ do

$d=\dfrac{\vec{r}_{j}-\vec{r}_{c}}{|\vec{r}_{j}-\vec{r}_{c}|}$

$\text{dist }\leftarrow\text{ d}$

end for

$d_{min}=\text{min(dist)}$

$\text{Get the index of }\mathcal{Q}\text{ for which }d_{min}=\text{
min(dist)}$

end for

Once the coordinates of all eight corner particles in the parallelepiped are
known, the estimation of the lattice parameters ($a_{d}$, $b_{d}$, $c_{d}$,
$\alpha_{d}$, $\beta_{d}$, $\gamma_{d}$) for the parallelepiped are carried
out. The lattice parameters consisting of the length of each vector and angles
between two unit vectors, are estimated from the joining vectors between each
of $N_{1}$, $N_{2}$ and $N_{3}$ particles with $N_{0}$ particle as the
identities of the corner particles of the parallelepiped are sealed. The
possible basis vectors of the parallelepiped are obtained by using equation 1
and 2 for a particular unit cell.

$\vec{\mathbf{a}_{d}}=\vec{r}_{1}-\vec{r}_{0}\text{ \&
}\vec{\mathbf{b}_{d}}=\vec{r}_{2}-\vec{r}_{0}\text{ \&
}\vec{\mathbf{c}_{d}}=\vec{r}_{3}-\vec{r}_{0}$ (1)
$\alpha_{d}=\cos^{-1}(\hat{\mathbf{b}}_{d}\cdot\hat{\mathbf{c}}_{d})\text{ \&
}\beta_{d}=\cos^{-1}(\hat{\mathbf{a}}_{d}\cdot\hat{\mathbf{c}}_{d})\text{ \&
}\gamma_{d}=\cos^{-1}(\hat{\mathbf{a}}_{d}\cdot\hat{\mathbf{b}}_{d})$ (2)

where $\vec{r}_{0}$, $\vec{r}_{1}$, $\vec{r}_{2}$ and $\vec{r}_{3}$ are the
position vectors of $N_{0}$, $N_{1}$, $N_{2}$ and $N_{3}$ respectively. These
vectors $\vec{\mathbf{a}_{d}}$, $\vec{\mathbf{b}_{d}}$, $\vec{\mathbf{c}_{d}}$
are appended into another set $\mathcal{P}_{all}$. So, each element of the set
$\mathcal{P}_{all}$ corresponds to a possible triplet of basis vectors for the
parallelepiped irrespective of satisfying any allowed Bravais unit cell or
not, under the detected class.

Figure 11: (A) The corner particles of the unit cell translated along the
three lattice vectors are shown in a cartoon diagram and (B) the unit cell of
the considered system with all the particles including the corners is shown.
The effective particle of an ideal $\beta$-Mn unit cell are evaluated and
shown in (C).

####  II.3.2 All particle coordinates of the parallelepiped

The identification of the coordinates of all particles is needed to know the
effective particles in the parallelepiped either it is a acceptable Bravais
lattice or not. Using a triplet (may or may not be the basis vectors), we
detect the eight corner particles of the parallelepiped. A convex-hull is
formed with the coordinates of eight corner particles which act like an
envelop. Considering the coordinates of all particles of the Bravais lattice,
computationally we search the particles which lie inside the constructed
convex hull using the Delaunay module of Scipy Virtanen  _et al._ (2020).
Following the method, we obtain all the particle coordinates staying inside
the parallelepiped or on the face of the parallelepiped as shown in Fig. 11B.
To handle the real data with noise, the hull is expanded isotropically by a
volume factor ($\mathcal{V}$), so that all the particles located at any sites
other than the corners, can be found inside the convex hull. For example, if
the initial or transformed Bravais system is face-centered lattice, the
particles exactly located on the faces of the parallelepiped (considered as
the unit cell box) can not be obtained due to the noise present in the system.
For that case, a little bit expansion of the convex hull is needed. For an
ideal crystal structure, $\mathcal{V}$ is set to 0. It is important to
remember that for the successful execution of this step, we need to use the
coordinates of the particles in the Bravais lattice only even if the initial
system is a non-Bravais one.

####  II.3.3 Coordinates of effective particles in a parallelepiped

After the identification of all particle coordinates in the parallelepiped
formed by a particular triplet, we can check whether the parallelepiped
represents a Bravais lattice or not, under the previously detected crystal
class. The coordinates of the effective particles in the parallelepiped are
needed to be detected to compare with the Bravais lattice.

Effective coordinates are related to one another by the crystallographic
symmetry properties. In principle, all the particle coordinates in the unit
cell can not be the effective particle coordinates. If the joining vector
between any two particles of the parallelepiped, $\vec{r}_{ij}$ does not
coincide with any of the vector directions i.e., $\hat{\mathbf{a}}_{d}$,
$\hat{\mathbf{b}}_{d}$, $\hat{\mathbf{c}}_{d}$ within the angle tolerance
$\mathcal{X}_{a}$ and the modulus of $\vec{r}_{ij}$ i.e., $|\vec{r}_{ij}|$
sufficiently differs from any of the $|\vec{\mathbf{a}}_{d}|$,
$|\vec{\mathbf{b}}_{d}|$, $|\vec{\mathbf{c}}_{d}|$ respectively barring the
distance tolerance $\mathcal{X}_{d}$, then the coordinate of the particle can
not be generated from the other one by translating any of the vectors within
the parallelepiped. So, two particles violating the condition are considered
as equivalent particles and only one particle is recognized as effective
particle instead of two. The algorithmic way of the determination of effective
particles is described bellow.

$\text{Coordinate of }N_{uc}\text{ particles }$

$\mathcal{D}=[]$

for $i\rightarrow(0\text{ , }N_{uc}-1)$ do

if $i\text{{ not in }}\mathcal{D}$ then

for $j\rightarrow((i+1)\text{ , }N_{uc})$ do

if $j\text{{ not in }}\mathcal{D}$ then

$\vec{r}_{ij}=r_{j}-r_{i}$

$\hat{r}_{ij}=\dfrac{\vec{r}_{ij}}{|\vec{r}_{ij}|}$

$\Theta_{1}=\cos^{-1}(\hat{r}_{ij}\cdot\hat{\mathbf{a}}),\Theta_{2}=\cos^{-1}(\hat{r}_{ij}\cdot\hat{\mathbf{b}}),\Theta_{3}=\cos^{-1}(\hat{r}_{ij}\cdot\hat{\mathbf{c}})$

if $\Theta_{1}\leq\mathcal{X}_{a}\text{ or
}\Theta_{2}\leq\mathcal{X}_{a}\text{ or }\Theta_{3}\leq\mathcal{X}_{a}$ then

if
$(|\vec{\mathbf{a}}|-\mathcal{X}_{d})\leq|\vec{r}_{ij}|\leq(|\vec{\mathbf{a}}|+\mathcal{X}_{d})\text{
or
}(|\vec{\mathbf{b}}|-\mathcal{X}_{d})\leq|\vec{r}_{ij}|\leq(|\vec{\mathbf{b}}|+\mathcal{X}_{d})\text{
or
}(|\vec{\mathbf{c}}|-\mathcal{X}_{d})\leq|\vec{r}_{ij}|\leq(|\vec{\mathbf{c}}|+\mathcal{X}_{d})$
then

$\mathcal{D}\leftarrow j$

end if

end if

end if

end for

end if

end for

$\mathcal{B}=[]$

for $i\rightarrow(0\text{ , }N_{uc})$ do

if $i\text{{ not in }}\mathcal{D}$ then

$\mathcal{B}\leftarrow i$

end if

end for

Figure 12: A flowchart showing the space group detection for all basis vectors
from the Bravais lattice

Where $N_{uc}$ is the total number of particles in the parallelepiped and
$\mathcal{B}$ is a list consisting the identities of the effective particle
only. After the identification of the coordinates of the effective particles
in a parallelepiped, we search for the unit cell of Bravais lattice under the
detected crystal class that matches with the parallelepiped. A Bravais lattice
is either a primitive cell or a “centered cell” with particles located at
either on the face mid-points, base mid-points or center of the parallelepiped
under a particular crystal class. Comparing the coordinates of effective
particles in the parallelepiped with all allowed Bravais lattices under the
crystal class (as shown in Fig. 11A) barring a tolerance value in real system,
the identification is carried out to make the decision about the chosen
triplet from $\mathcal{P}_{all}$ to be considered as basis vectors or not.
Each possible triplet of the set $\mathcal{P}_{all}$ is chosen and checked to
be recognized as a valid choice of a permitted Bravais unit cell under the
detected crystal class. In this way, all triplets satisfying the condition are
identified and considered as a set $\mathcal{P}$, where total number of such
triplets is $n_{p}$. As each of these triplets satisfies the requirement of an
allowed Bravais unit cell, a triplet can be accepted as $\vec{\mathbf{a}}$,
$\vec{\mathbf{b}}$, $\vec{\mathbf{c}}$. A schematic diagram is presented in
Fig. 12 describing the protocol of identification of the basis vectors for the
structure. After identifying the allowed triplets for the basis vectors, the
next step is to construct the unit cell of the initially considered structure
which is discussed below.

###  II.4 Unit cell formation of the crystal structure

Each triplet of the set $\mathcal{P}$ is considered that can span an allowed
unit cell to generate the entire Bravais lattice. We detect only the corner
particles of the Bravais unit cell for each triplet chosen from the set
$\mathcal{P}$ by searching the particles at the nearest distances followed by
the construction of a convex hull as described earlier. Now all the particle
coordinates of the initially considered system (a non-Bravais lattice if
transformed) are taken into account and we find the coordinates of all
particles in the unit cell by expanding the convex hull to handle the noise of
the system, following the same protocol discussed in the above section. The
unit cell for the considered crystal structure with all particle coordinates
and basis vectors is formed leading to the determination of lattice parameters
of the unit cell ($a$, $b$, $c$, $\alpha$, $\beta$, $\gamma$). We also detect
the coordinates of effective particles in the unit cell following the similar
approach as discussed. Now all the necessary information of the unit cell is
known for a particular choice of basis vectors and all the triplets of the set
$\mathcal{P}$ are allowed for the structure.

The choice of the lattice parameters of the determined unit cell for
$\beta$-Mn crystal structure was unique and the values were $a$=6.3150024,
$b$=6.3150024, $c$=6.3150024, $\alpha$=90∘, $\beta$=90∘, $\gamma$=90∘, which
exactly matched with the lattice parameters initially used to construct the
unit cell Xie  _et al._ (2013). The protocol confirmed that the unit cell of
$\beta$-Mn crystal structure consisted of 20 effective particles and the
coordinates were known.

###  II.5 Determination of space group of the crystal structure

As the final goal, we determine the space group of the considered crystal
structure for a particular choice of basis vectors with the complete knowledge
of the unit cell. For this, Spglib package is used to get all the
crystallographic symmetry operations of the corresponding crystal structure
and finally the space group. Computationally, we check for all the possible
symmetry operations of the unit cell i.e. translation, rotation, reflection,
screw axes and glide planes. This leads to the determination of the space
group properly with a particular set of triplet for which the possible
crystallographic symmetry operations are obtained and applied correctly. The
three dimensional Cartesian coordinates of the effective particles are
transformed into fractional coordinates and the lattice parameters along with
those fractional coordinates are used as inputs. All the possible
crystallographic symmetry operations were checked computationally by the
package within the tolerances (for ideal system, the tolerances are zero) and
the space group of the crystal structure is obtained as outputs according to
the standard nomenclature IuC (2016). Space group detection is quite
straightforward for an ideal crystal and it is complicated for a noisy
crystal, as we need to tune two tolerance values, symprec, i.e. the distance
tolerance in Cartesian coordinates and angle_tolerance to determine the space
group of the crystal structure as well as the all possible crystallographic
symmetry operations. Unsuccessful determination of any symmetry operation can
affect the space group leading to the determination of an incorrect one; so it
is very sensitive on the tolerances. Corresponding crystallographic symmetry
operations can also be obtained using the get_symmetry module of the package
specifying the lattice and basis information in proper format. The same space
group is determined for all $n_{p}$ choices of basis vectors of set
$\mathcal{P}$. Multiple choices in the real-space do not affect the space
group as a unique choice of fractional coordinates defining the symmetry of
the crystal is obtained from any particular set of the effective particle
coordinates and basis vectors. In general, the number of choices for basis
vectors depends on the considered crystal structure. The algorithmic recipe
not only provides the complete information of the unit cell and space group of
the crystal structure but also the possible choices of the valid basis vectors
are obtained. Considering the coordinates of the particles as the only inputs,
the protocol provides the unit cell information and corresponding space group
irrespective of any complex structure as shown by the schematic diagram in
Fig. 13. The detail implementation is showcased step-by-step for the ideal
$\beta$-Mn crystal structure in Fig. 14.

Figure 13: The complete summary of the approach is illustrated in this figure.
Figure 14: (A) An ideal beta-Mn crystal structure is considered followed by
the determination of radial distribution function (B), bond-orientational
diagram (C) and the implementation of the K-Means clustering algorithm (D). As
a part of the environment separation technique, the particles are colored
based on the similar kind of environment and shown in (E). The particles
sharing the similar kind of environment (particles with “blue” color only) are
separated out as shown in (F). The radial distribution function (G), bond-
orientational order diagram (H) and the K-Means algorithm (I) are implemented
on the transformed Bravais lattice system. The unit cell of the system and
effective particle are shown in (J) and (K) respectively.

The space group of ideal $\beta$-Mn crystal appeared to be $P4_{1}32$
according to the international symbol as detected by the method where only one
choice of the basis vectors ($n_{p}$ = 1) was observed. Each step of the
detection of the unit cell of beta-Mn crystal structure is shown in Fig. 14.
Further examples of simple and complex crystal structures will be discussed
along with an example of a system directly obtained from simulation data. The
results will mainly focus on the implementation of the scheme in different
kinds of crystal structures as the validation purposes.

## Results and Discussion

In this research, the data of three synthetically prepared ideal crystalline
systems are reported by replicating the corresponding unit cells
(Crystallographic Information File, i.e., CIF files are available in
Crystallographic Open Database (COD) system Gražulis  _et al._ (2012)) in
three-dimensions where the lattice parameters and space group (as denoted by
International symbol) are known. We applied the protocol and checked whether
the same unit cell and space group can be obtained from the bulk crystal
structure using the coordinates only. The detected unit cells of these systems
along with all steps are shown below including the data of simulated cubic
diamond crystal structure by varying the noise levels. Here, we demonstrate
our approach using four crystal structures, (i) an ideal body-centred cubic
(BCC) crystalline system with space group $Im\bar{3}m$, (ii) an ideal system
of C3O6Sr monoclinic crystal with space group $P_{1}2_{1}/C_{1}$, (iii) a
synthetically prepared Hexagonal-closed packed (HCP) structure with space
group $P6_{3}/mmc$ and (iv) the simulated cubic diamond crystal structures
with space group $Fd\bar{3}m$ and eight effective particles in the unit cell
at different noises. The last example depicts the system with the different
statistical noises as those comes out of computer simulations. The scheme is
applied step by step on each system and the detected unit cells, space groups
are shown as results.

Figure 15: (A) An ideal BCC crystal of Ba is shown along with the unit cell in
the inset. RDF and the distribution of the first nearest-neighbors are shown
in (B) and (C) respectively with $r_{c}$ $\sim$ 4.5. There are eight clusters
as confirmed by elbow analysis of the K-Means clustering (D). The BCC unit
cell contains total nine particles as shown in (E).

Fig. 15A shows an ideal BCC crystal, manually generated from the unit cell of
Barium (Ba) with lattice parameters $a$=4.758, $b$=4.758, $c$=4.758,
$\alpha$=90∘, $\beta$=90∘, $\gamma$=90∘ Kitano  _et al._ (2001) (inset of
(A)), followed by a calculation of RDF of the system, showing crystal-like
nature (Fig. 15B). The nearest neighbor distance ($r_{c}$) is calculated from
the first minima of the RDF which turned out to be $\sim$ 4.5 (exact nearest
neighbor distance was 4.12). Further, considering all pairwise distances, the
positional distributions of the neighbors within $r_{c}$ is shown in (Fig.
15C), formed eight particles (zero positional deviation for the ideal system)
in three-dimensional space, as confirmed by the elbow analysis of K-Means
clustering algorithm as in Fig. 15D. The environment separation was performed
and it turned out that $\sim$ 100% particles had similar kind of environment
confirming the system as a Bravais lattice already. For this example, the
transformation from a non-Bravais lattice into a Bravais lattice was not
required. The coordinates of the centroid of the clusters (here the coordinate
of the particles in the BOO) were considered as the vertices of a convex
polyhedron. After the decomposition of the convex polyhedron, the vertices
forming the polyhedron’s faces and edges were evaluated followed by the
calculations of determining the coordinates of the six face mid-points and
twelve edge mid-points. Total number of non-unit vectors ($g$) calculated from
the polyhedron was 26. Each triplet of ${}^{26}C_{3}$ combinations was treated
as a set of lattice vectors ($\vec{\mathbf{a}}_{d}$, $\vec{\mathbf{b}}_{d}$
and $\vec{\mathbf{c}}_{d}$) and the lattice parameters, $a$, $b$, $c$,
$\alpha$, $\beta$ and $\gamma$ were determined. Computationally we checked
each set of lattice parameter to know the crystal class satisfied by the
those. Cubic crystal class was satisfied by few triplets as the highest order
of symmetry i.e., 48. The directions of basis vectors appeared to be unique
satisfying the cubic class and $\hat{\mathbf{a}}$ = [1, 0, 0],
$\hat{\mathbf{b}}$ = [0, 1, 0] and $\hat{\mathbf{c}}$ = [0, 0, 1]. No other
directions produced any valid Bravais lattices under the cubic class.
Construction of the unit cell with the basis vectors followed the
determination of actual lattice vectors, $\vec{\mathbf{a}}$ = [4.758, 0, 0],
$\vec{\mathbf{b}}$ = [0, 4.758, 0] and $\vec{\mathbf{c}}$ = [0, 0, 4.758] with
lattice parameters 4.758, 4.758, 4.758, 90∘, 90∘, 90∘. The BCC unit cell with
eight particles at corners and one particle at the centre of the cubic box was
obtained as shown in Fig. 15E. The two effective particles were obtained and
the space group detection technique was applied with four particle basis and
the space group turned out to be $Im\bar{3}m$ considering symprec = 0.0 and
angle_tolerance = 0∘.

Figure 16: Environment separation technique was carried out on the system of
C3O6Sr crystal showing the transformation of the complex non-Bravais structure
into a “simple cubic” Bravais lattice at two steps. The initial non-Bravais
lattice is shown in (A). After categorizing based on the environments, the
particles are shown in different colors, where the particles with any single
color have the similar kind of environment (B). The configuration with only
blue color was chosen as system (C) followed by the categorization of the
particles based on the the environments as shown in (D). Then the
transformation of the Bravais lattice successfully happened after separating
out the particles with the “pink” color only from the system (D) as shown in
(E). Figure 17: (A) An ideal multi-component monoclinic crystal of
$C_{3}O_{6}Sr$ is shown along with the unit cell (inset). The effective
coordinate of the particles irrespective of the size or particle types are
shown in (B), followed by the corresponding RDF and distribution of first
nearest-neighbor in (C) and (D) respectively. Number of clouds with zero
positional deviation is four as confirmed by the elbow analysis of K-Means
clustering (E). After environment separation for five times, using the
particles sharing the same kind of environment, the system snapshot is shown
in (F) RDF is calculated (G). Choosing $r_{c}$ $\sim$ 9.5, the positions of
the local neighbors are illustrated in (H). There exists eight unique clusters
according to (I) and the monoclinic unit cell is shown in (J).

Another ideal atomic crystal of C3O6Sr (space group - $P_{1}2_{1}/C_{1}$ )
with complex basis was prepared with lattice parameters, $a$ = 7.966, $b$ =
9.205, $c$ = 7.319, $\alpha$ = 90∘, $\beta$ = 102.104∘ and $\gamma$ = 90∘
Vanhoyland  _et al._ (2001). Fig. 17A exhibits the synthetically prepared
crystal structure by replicating the unit cell in the three dimensions, where
the unit cell is shown in the inset of Fig. 17A. Only the coordinates of each
kind of atoms were considered irrespective of their sizes or atom types and
the configuration is shown in Fig. 17B. We intentionally loose all the
information except the coordinates of all particles for the validation
purpose. The RDF was calculated and the value of $r_{c}$ was chosen as 1.2.
The positional distribution of nearest neighbors within $r_{c}$ is shown in
Fig. 17D. Only four clusters were detected as confirmed by the elbow analysis
after the K-Means clustering (Fig. 17E) was done. The environment separation
technique was carried out two times in the iterative way, until each particle
had similar kind of positional environment in the transformed system, i.e. the
non-Bravais lattice turned into a Bravais lattice. The configurations of the
two steps are shown in Fig. 16, with the categorization of the particles based
on the cluster ids in the corresponding BOOs. After the transformation, the
system snapshot is shown in Fig. 17F. Hence the “pink” color of the particles
in the Cubic Bravais lattice (shown in Fig. 16E) was changed into “blue” color
(Fig. 17F) for the visualization purposes only, where the coordinates of the
particles in the two systems shown at Figs. 16E,17F were kept intact except
for the color. The RDF was calculated further and shown in Fig. 17G and
$r_{c}$ was chosen as 9.99, confirming the distribution of the positions of
the nearest neighbors in three-dimensional coordinate system as shown in Fig.
17G. Eight distinct clusters (for ideal system, only particles) were found
(Fig. 17H), which is confirmed by the elbow analysis plot of K-Means
clustering (Fig. 17I). Construction of the convex polyhedron provided the
directions of fifteen possible vectors, but all those triplets did not
replicate the valid unit cell under Monoclinic crystal class. We separated out
the basis vectors corresponding to the primitive Monoclinic unit cell with one
effective particle and there were eight unique choices. A single triplet from
the set of eight choices was considered and the unit cell of the crystal
structure was detected. One of the valid choices of basis vectors was
$\vec{\mathbf{a}}$ = [-2.448, 0, -7.157], $\vec{\mathbf{b}}$ = [-4.750, 0,
3.578] and $\vec{\mathbf{c}}$ = [0.0, -9.205, 0.0] and the lattice parameters
were 7.966, 9.205, 7.319, 90∘, 102.104∘, 90∘ along with other seven possible
choices. For the validation purpose, the unit cell of the crystal structure
was generated by translating along the directions of that basis vectors. After
obtaining eight corner particles of the Monoclinic cell, the convex hull was
constructed followed by the detection of the coordinates of all particles in
the unit cell for the initial non-Bravais lattice system. As this was an ideal
structure without noise, the expansion parameter of the convex hull
$\mathcal{V}$ was set to zero. A randomly chosen unit cell is shown in Fig.
17J, which confirms the lattice parameters to be exactly same as our initial
consideration where all other choices of different lattice parameters were
present with different basis vectors. The unit cell fails to provide any
atomic information of the basis upon the execution of the algorithm as we have
used the coordinates only by discarding all other details. The space group
appeared to be $P_{1}2_{1}/C_{1}$ for each of the eight choices, which
validates the success of the methodology towards the determination of unit
cell of the crystal structure in the real-space.

Figure 18: Transformation of a complex Hexagonal-closed packed (HCP) structure
into a Bravais lattice is shown using the “environment separation” technique.
(A) The particle coordinates in a synthetically prepared ideal HCP crystal
structure are shown. After categorizing the particles based on the
environments, the configuration is shown (B), where particles with same color
have similar kind of environment in the configuration. (C) Finally the
particles with any color (here “blue” color) are chosen and the system
satisfies the criteria of a Bravais lattice. Figure 19: The steps are shown
for a synthetically prepared HCP crystal structure. (A) The coordinates of the
particles are shown followed by the calculation of RDF (B), BOO (C) and elbow
analysis after K-Means clustering (D). After separating the particles with
similar environment for two times as shown in the Fig. 18, the transformed
Bravais system is shown in (E). Considering the Bravais lattice, again the RDF
(F), BOO (G) and the elbow analysis (H) are calculated. All particles in the
unit cell and effective particles of the HCP structure are shown in (I) and
(J) respectively.

The next example is the detection of the unit cell and a space group of a
synthetically prepared ideal Hexagonal-closed packed (HCP) structure with
lattice parameters 5.947, 5.947, 8.375, $90^{\circ}$, $90^{\circ}$,
$120^{\circ}$ and space group $P6_{3}/mmc$, having 22 effective particles in
the unit cell. Our aim was to get back the exact lattice parameters and the
space group following the defined approach, as used to construct the HCP
structure. The coordinates of the particles were considered from the HCP
crystal structure of K2SO4 by loosing all other information intentionally
expect the coordinates of the atoms, already reported in the literature
Miyake, Morikawa, and Iwai (1980). The steps to separate the particles with
similar kind of environment are illustrated in Fig. 18. The coordinates of the
particles of a HCP structure are shown in Fig. 18A. As the first step of the
layout, we checked whether the considered system was a Bravais lattice or not.
To check this, the “environment separation” technique was carried out and the
particles were categorized based on the unique environments. The configuration
shown in Fig. 18B suggested that the considered system was non-Bravais in
nature due to the existence of multiple kind of environments, as the particles
with multiple colors existed. As the part of the “environment separation”
technique, the particles with any single color (for this example, “blue”
color) were chosen (Fig. 18C) and considered further to check that whether the
extracted system (Fig. 18C) was Bravais or not. The configuration shown in
Fig. 18C turned out to be a hexagonal Bravais lattice, satisfying all the
particles with similar kind of environment. For this structure the
transformation of a non-Bravais lattice into a Bravais lattice occurred in a
single iterations. The other steps of the scheme are described in Fig. 19. The
coordinates of the particles in the HCP structure were considered as the
initial system followed by a calculation of the RDF (Fig. 19B) where $r_{c}$
$\sim$ 2.0. The three dimensional distributions of the local neighbors within
the distance $r_{c}$ included 28 clusters (here only the points due to the
ideal structure) as confirmed by the elbow analysis after the K-Means
clustering was done (Fig. 19D). After separation of the environments (as
described in Fig. 18), the hexagonal Bravais lattice was obtained and shown in
Fig. 19E. The two configurations in Fig. 18E and Fig. 19E are exactly same.
Considering the Bravais lattice, the RDF was calculated again with $r_{c}$
chosen at 6.0 (Fig. 19F). The positional distribution of the local neighbors
of all particles within the distance (Fig. 19G) suggests the existence of
eight clusters (here only points) as assured by the elbow analysis shown in
Fig. 19H. After the construction of convex polyhedron, all possible directions
of the vectors, $\hat{\mathbf{a}}_{d}$, $\hat{\mathbf{b}}_{d}$,
$\hat{\mathbf{c}}_{d}$ were found. The valid choice of the directions of the
primitive vectors appeared to be unique for this primitive Hexagonal Bravais
lattice. Considering a reference particle from the Bravais lattice, the other
seven corner particles were found by translating along each of the three
primitive vector directions based on the minimum distances. Here both the
tolerances, $\mathcal{X}_{d}$ and $\mathcal{X}_{a}$ were set to zero as there
was no noise in the system. The lattice parameters calculated using the
protocol were 5.947, 5.947, 8.375, $90^{\circ}$, $90^{\circ}$, $120^{\circ}$,
satisfying the Hexagonal crystal class. The exact lattice vectors for the
system were $\vec{\mathbf{a}}$ = [5.947, 0, 0], $\vec{\mathbf{b}}$ = [2.973,
5.15, 0] and $\vec{\mathbf{c}}$ = [0, 0, 8.375]. After the formation of the
convex hull with eight corner particles, all other particles except the eight
corner ones, participating in the unit cell were found and shown in
$\mathcal{X}_{d}$I. The coordinates of the effective particles were searched
(shown in $\mathcal{X}_{d}$J) in order to calculate the space group and the
space group turned out to be $P6_{3}/mmc$. For this example also, the defined
methodology performed well and detected the unit cell as well as the space
group of the crystal structure accurately.

Figure 20: (A) A system of cubic diamond crystal structure (complex basis) is
shown along with the determination of corresponding RDF (B), the positional
distribution of all neighbors in the system (BOO) (C) and elbow analysis of
the K-Means clustering (D). After environment separation, the particles are
shown in different colors in (E) based on the environments showing majority of
the particles to have two distinct colors, “blue” and “green”. The rest of the
colors are in very less in number; those can be treated as noise. The
coordinates of the particles with only one color (“blue”) was separated out
and considered as the system as shown in (F) followed by the determination of
the RDF (G), BOO (H) and elbow analysis after the K-Means clustering (I).
Finally the unit cell of the cubic diamond structure and eight effective
particles in the unit cell are shown in (J) and (K) respectively. Figure 21:
All the steps are showcased for a system of the cubic diamond crystal with
higher noise level than the system shown in Fig. 20. The numbers of the
subfigures follow the same as defined in Fig. 20. Figure 22: All the steps are
shown for a much higher noisy system of the cubic diamond crystal than the
system shown in Fig. 20. The numbers of the subfigures follow the same as
defined in Fig. 20.

The last example corresponds to the cubic diamond structure with complex
basis, coming out of simulation Zhang  _et al._ (2005); Dshemuchadse, Jung,
and Steurer (2011); Wang  _et al._ (2017); Damasceno, Engel, and Glotzer
(2012); Klotsa  _et al._ (2018); Marson  _et al._ (2019). The configuration is
shown in Fig. 20A along with the RDF, BOO and elbow analysis after the K-Means
clustering as shown in Fig. 20B,C,D. For this structure, $r_{c}$ was chosen as
1.1 as the first minima of the RDF distribution. Unlike the previous ideal
systems, because of the statistical noise, total eight clusters of particles
with significant positional deviations were observed followed by a
confirmation of eight clusters by the elbow analysis of K-Means clustering.
The environment separation technique was carried out on the system to
categorize the particles based on the environments around themselves. The
configuration shown in Fig. 20E showed the majority of the particles in mainly
two colors, “blue” and “green”, where the rest of the colors could be treated
as noise due to the very less occurrence of such particles. By checking the
environment, $\sim$ 48.73% particles of the initial considered system appeared
to possess the similar kind of environment and was shown in Fig. 20E. As the
next step, the particles having similar kind of environment, i.e., the
particles with similar color (let’s say, “blue” color) were separated out and
the coordinates of those particles were treated as the system of consideration
for performing the next few steps. The particles in the “black” color as shown
in the configuration (Fig. 20F) can be considered as noise. For this case the
environment was separated within one iteration resulting $\sim$ 93% particles
in the updated system had the similar kind of environment, satisfying the
criteria of the Bravais lattice in the presence of the noise. Unlike te
previous systems with the Simple Cubic (SC) Bravais lattice, here the
transformed Bravais unit cell appeared to be FCC in nature. The RDF, BOO and
K-Means clustering were analysed for the Bravais lattice exhibiting standard
signature of FCC crystal Fig. 20G,H,I; where $r_{c}$ was set at $\sim$ 1.75.
In the transformed FCC crystal, twelve distinct clusters were observed as
assured by the elbow analysis of K-Means clustering. Consideration of the
centroid of the twelve clusters as the vertices of a convex polyhedron, the
coordinates of 14 face mid-points and 24 edge mid-points provided a total of
50 non-unit directions as designed in the protocol. All possible combinations
of ${}^{50}C_{3}$ were calculated as we got a set of possible triplets. We
checked each triplet to satisfy the possible crystal classes within the
tolerance values. The crystal class appeared to be cubic as it turned out to
be of the highest order satisfied by a particular triplet. The choice of the
primitive vectors appeared to be unique for simple cubic unit cell which were
$\vec{\mathbf{a}}$ = [-0.395, 0.212, -0.940], $\vec{\mathbf{b}}$ = [-0.534,
0.811, 0.383] and $\vec{\mathbf{c}}$ = [-0.791, -0.646, 0.216] which did not
commensurate with the simulation frame. Choosing one random particle from the
system and associating it with a lattice site, we translated along the
directions of primitive lattice vectors and got all the corner particles of
the unit cell. After the construction of the convex hull using eight corner
particles of the transformed Bravais lattice, the hull was expanded a little
bit to handle the noise of the system, i.e. $\mathcal{V}$ = 0.1 times multiple
of the total volume of the hull. All the particles in a randomly chosen cubic
diamond unit cell are shown in Fig. 20J. Constructing the unit cell, the
actual lattice vectors were evaluated followed by the determination of the
lattice parameters; $a$= 2.101, $b$=2.107, $c$=2.16, $\alpha$ = 89.87∘,
$\beta$=89.30∘ and $\gamma$=88.88∘. The effective particles in the unit cell
are shown in Fig. 20K and the space group was also checked for the unit cell
of the cubic diamond structure, which appeared to be $Fd\bar{3}m$ with the
consideration of two tolerances; symprec = 0.1 and angle_tolerance = 6∘.

The same strategy was applied on the cubic diamond crystal structures
considering two different level of noises, to test the validity of the method.
The systems shown in the Figs. 21, 22 correspond to the cubic diamond
structure simulated at two lower packing fractions than the previous one
(shown in Fig, 20). Bond-order diagrams (Fig. 20C, 21C, 22C) could be used as
the standard visual descriptor of statistical noise, which suggest that the
system shown in Fig. 22 is more noisy than the other two. The Figs. 21, 22
suggest the robustness of the methodology satisfying the same type of unit
cell with eight effective particles and same fractional coordinates in the
reference frame of the corresponding unit cell of the cubic diamond structure.
Only the unit cells got distorted little bit depending on the noise present in
the crystal structures. Except the fact, all the steps exhibited the same
qualitative behavior as showcased in Figs. 21, 22. The positional deviation of
the points in the distribution of the local neighbors (Fig. 21C and 22C) are
much higher as compared to the same of Fig. 20C resulting the same transformed
FCC crystals each time. Only the distance and angle tolerances used to handle
the noisy crystal structures were little bit higher, which completely made
sense. The tolerances were $\mathcal{X}_{d}$ = 0.15, $\mathcal{X}_{a}$ = 10∘
and $\mathcal{X}_{d}$ = 0.2, $\mathcal{X}_{a}$ = 15∘ for the noisy crystals
shown in Figs. 21, 22 respectively. The same space group $Fd\bar{3}m$ appeared
for each of the cases indicating the robustness of the approach.

##  III Conclusions

We developed a direct approach to detect the unit cell and space group of the
crystal structure based on the real-space analysis. Despite the coordinates of
the particles, the analyses of RDF and BOO lack the precise information of the
crystal structure. The prescribed route involves these steps which directly
depend on the particle coordinates in the real space. We also use the
positional distribution of the local neighbors to identify the directions of
lattice vectors which is the most important and challenging step in this
route. In order to tackle the problem, one fundamental step is to find out the
crystal class among seven classes in three dimensions after decomposing a
complex structure into a simple Bravais lattice. We develop a robust
computational way based on the difference between the Bravais and non-Bravais
lattice allowing the transformation of any complex non-Bravais crystal
structure into one of the fourteen Bravais lattices in the three dimensions.
We use the idea of effective particles depending on the distributions of the
neighbors and the transformation of a complex crystal into a simple Bravais
lattice allow us to detect the directions of lattice vectors in a much simpler
way. Based on the observations we note that each direction of the lattice
vectors intersect any of the vertices, face mid-points or edge mid-points of
the convex polyhedron constructed from the BOO. The computational approach
using the clustering method is used to deal with the noise present in the
system. The identification of the primitive lattice vector directions from the
simulated system is very crucial where the primitive lattice vectors do not
commensurate with the simulation frame. This approach makes our goal more
achievable in the context of the detection of the unit cell as the directions
of the broken symmetries were identified. It is quite straightforward that for
any structures with primitive unit cell, the directions of the primitive
lattice vectors pass though any of the vertices of the constructed polyhedron.
Upon the detection of the lattice vectors and coordinates of the effective
particles all the crystallographic symmetry operations of the corresponding
unit cell were obtained. Successful execution of the recipe determines the
unit cell and space group of the crystal structure, which is reported in this
article for the ideal systems and the simulated systems with different noise
levels.

The rotational invariance property of the crystal structure, defined by
Steinhardt et al. is widely used as a descriptors for the identification of
FCC, BCC, HCP structures Steinhardt, Nelson, and Ronchetti (1983), while
Ackland-Jones order parameter also distinguishes the local environment of FCC,
BCC and HCP Ackland and Jones . These order parameters are also useful to
distinguish the solid or liquid like particles. The usages are limited to a
few specific crystal structures and lack the general approach to detect any
crystal structures. The reported scheme can not be treated as a direct
extension or one to one correspondence with either of former two approaches,
commonly used in the context of crystal structures detection. The polyhedral
template matching method was reported to identify the local environment of a
few crystal structures but this method does not work for any crystal
structures specially for the structures with crystal classes of lower
symmetries such as orthorhombic, monoclinic, triclinic due to the variation of
the distributions of the local neighbors Larsen, Schmidt, and SchiØtz (2016).
Unlike the blueprint discussed in this article, the former approaches were not
capable to provide the complete information of the unit cell along with the
space group, in the presence of translational broken symmetry state satisfying
the global rotational invariance. We did not incorporate the concept of
“Wyckoff positions” in the strategy, which is one of the most common
attributes used to identify the crystal structures Kittel (2005); Massa
(2004). On the other hand, in the modern machine learning based approach, a
lot of reported structures are required as footprints for purely training
purposes with a well defined and general descriptor. Our direct approach
promises to detect any single crystal structures including the structure with
complex basis and takes minimal time to detect the crystal structure
accurately.

As the limitations of the discussed policy, this can not be used to study the
process of crystal nucleation which requires a major modifications to the
approach. This particular approach needs the assurance of the structure to
have the complete translationally broken symmetry state in three dimensions as
it does not check whether the considered structure is a crystal or not. Again
this will not be useful to study the poly-crystalline structure; the
modifications or extensions are kept for further studies. The detection of
space group for the chiral crystal structure may fail or yields incorrect
result due to the inappropriate implementation of the “environment separation”
protocol which requires to be modified according to the chirality Fecher,
Kübler, and Felser (2022). If the simulated crystal structure is too much
noisy, one will definitely find difficulties to detect the unit cell of the
crystal as the method fails to find the exact number of the clusters formed by
the neighbors. Despite the dependency of the difficulty level on the noise
present in the system, the methodology is robust, exact and converges. Only
three dimensional crystal structures can be handled following the protocol;
the implementation in any other dimensions are kept for further
investigations. The method depends on some tolerance values, which strictly
depend on the noise present in the crystal structures. Despite a few
restrictions in the context of the application of computational layout, this
promising approach is very useful to detect the crystal structures from real-
space data in the field of condensed matter physics, applied physics or
material sciences.

###### Acknowledgements.

We acknowledge financial support from DST-INSPIRE Fellowship (IVR No.
201800024677) provided to SK. AD thanks DST-SERB Ramanujan Fellowship
(SB/S2/RJN-129/2016) and IACS start-up grant. KC thanks IACS for financial
support. Computational resources were provided by IACS HPC cluster and partial
use of equipment procured under SERB CORE Grant No. CRG/2019/006418.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

SK and AD designed the research. SK and KC executed the idea computationally
and implemented on multiple systems. SK and AD wrote the paper.

## DATA AVAILABILITY

The data that support the computational approach of this method are available
within the article.

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