# Identification of Local Structure in 2-D and 3-D Atomic Systems through Crystallographic Analysis

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Characteristic Crystallographic Element Norm

_{at}spherical particles or atoms, where each particle i ($i\in \left[1,\dots ,{N}_{\mathrm{at}}\right]$) is defined by its position vector,

**r**

_{i}.

_{coord}(X) taking the values of 3, 4, 5, 6, 8 and 12 for HON, SQU, PEN, TRI, HEX/BCC and HCP/FCC/FIV, respectively. Figure 3 and Figure 4 present the reference atom, the closest neighbors as well as the corresponding Voronoi polyhedron (cell) for the reference crystals in 3-D (HCP, FCC, HEX and BCC) and 2-D (TRI, SQU and HON), respectively. In the 3-D case, atoms are shown with reduced dimensions for visualization purposes.

_{coord}, i.e., if the number of Voronoi neighbors is smaller than the coordination number of the reference crystal X, a penalty function is introduced in the form:

^{0}is a constant whose value is equal to 0.07. This value is empirically determined so that the CCE can differentiate crystals that belong to the same point group but have different number of neighbors and Voronoi polyhedra (for example BCC vs. FCC), as will be explained in the Results section. In computer-generated systems the local environment does not correspond to the Voronoi polyhedron of a perfect crystal due to fluctuations in atom positions. Such typical Voronoi cell has a higher number of vertices and faces as some of the second-nearest neighbors lie closer to the central, reference atom than those of a perfect crystal, i.e., N(i) > N

_{coord}(X). In such cases, the penalty function is zero and the CCE proceeds by sorting the neighbors based on their distance from the reference atom and selecting only the N

_{coord}(X) closest ones.

_{coord}(X)), the geometric symmetry elements, N

_{el}(X), and the corresponding actions for each element, N

_{act}(k,X) ($k\in \left[1,\dots ,{N}_{\mathrm{el}}\left(X\right)\right])$ for each X crystal in 3-D and 2-D, are presented in Table 1 and Table 2, respectively. The analogous combinations for the fivefold (FIV) local symmetry in 3-D and the pentagonal (PEN) local symmetry in 2-D are also included.

**r**

_{j}, considered relative to the ones of the reference atom, against the “ideal” coordinates ${R}_{j}^{X}$ ($j\in \left[1,\dots {N}_{\mathrm{coord}}\left(X\right)\right]$) of the sites that constitute the coordination polyhedron of the perfectly ordered structure X. In the most general form the CCE norm can be defined as:

_{coord}(X) sites, geometric elements N

_{el}(X) and N

_{act}(k,X) symmetry actions corresponding to the reference crystal X. ${S}_{k,m}^{X}$ is the orthogonal matrix that performs the m

_{th}action of the k

_{th}element. For example, the corresponding matrix for the point inversion for the FCC, HEX and BCC crystals is simply:

**S**

_{1}is found by scanning the whole spherical domain (θ

_{1}, φ

_{1}); given the

**S**

_{1}orientation, the second axis is built with an angle of θ

_{2}= 70.53

^{Ŷ}with respect to the first, and a search is now carried out in the azimuthal space: φ

_{2}($\in \left[0,2\pi /3\right]$). Given

**S**

_{1}and

**S**

_{2}the third (

**S**

_{3}) and fourth (

**S**

_{4}) symmetry axes are fully defined and no further search must be carried out. Each time the geometric element is defined, the analogous symmetry actions (roto-inversions for FCC and BCC) are performed over all coordination neighbors according to Equation (2); as a result, each possible axis orientation, or their combination in case of more than one, provides a different norm value. For reference site, i, and given ordered structure X, the final CCE norm, ${\epsilon}_{i}^{X}$, is set to be the minimum of this iteratively refined set of values. By construction, the closer the norm to zero the higher the similarity of the local environment to the ideal X crystal. Moreover, by definition the X-CCE norm of the X reference crystal should be zero, subject to the statistical error imposed by the size of the scanning grid imposed on the spherical domain (see below). Furthermore, by utilizing the characteristic elements and actions as the fingerprint of X-crystal the CCE algorithm is highly discriminating: for two different crystals X and Y, if ${\epsilon}_{i}^{X}\to 0$ then ${\epsilon}_{i}^{Y}>0$ and vice versa.

^{thres}which is the same for all reference crystals. If the CCE norm is lower than this threshold (i.e., ${\epsilon}_{i}^{X}$< ε

^{thres}) the reference i site has similarity to that crystal and is thus labelled as X-like. Past studies on bulk, 3-D systems consisting of non-overlapping [29,30,31,34,35,36] or fused [38,39] spheres adopted thresholds of 0.245 and 0.200, respectively. The threshold value is identified through parity (${\epsilon}_{i}^{X}vs.{\epsilon}_{i}^{Y}$) plots over all particles in the system as the ones to be presented below. As it will be demonstrated in the results section the same threshold (ε

^{thres}= 0.245) can be adopted in the case of hard spheres in 2-D films.

^{X}, can be calculated:

_{s,d}corresponds to the number of such crystal templates. In the present demonstration of the CCE norm N

_{s,3}= 4 (HCP, FCC, HEX and BCC) and N

_{s,2}= 3 (TRI, SQU, HON).

#### 2.2. Molecular Simulations

_{ij}, according to the hard sphere (HS) model the energy is either zero or infinite:

_{1}is the collision diameter, also taken as the characteristic length of the system. The second model is the square well (SW), which further includes an attractive range (if ${r}_{ij}\in \left[{\sigma}_{1},{\sigma}_{2}\right]$) with a coresponding intensity, ${\epsilon}_{SW}$:

_{wall}, is equal (within a tolerance of 10

^{−4}) to the sphere diameter, σ

_{1}. Once this situation is met the system is practically converted into a 2-D, thin film whose thickness corresponds to a single layer (d

_{wall}→ σ

_{1}). Two different sets of simulations were carried out containing 100 chains of average length N

_{av}= 12 and 50 chains of N

_{av}= 24. Due to the application of chain-connectivity-altering moves dispersity is introduced in chain lengths, which fluctuate uniformly in the intervals N

_{ich}$\in $ [6,18] and [12,36] for N

_{av}= 12 and 24, respectively.

_{B}, k

_{B}being the Boltzmann constant); (ii) all bonds are eliminated so that the system consists now of monomers and the SW potential is further activated. For the polymer-based systems, the equilibration algorithm and the corresponding MC moves are similar to the ones used for HS [30,33,34,48,49,50,51,52,53] and SW [41] chains. For the monomer-based systems two moves are employed: simple sphere displacements whose amplitude is auto-configured based on the acceptance rate and cluster-based displacements following the cluster detection/translation algorithm presented in Ref. [41]. Simulations on 3-D SW polymers and monomers are conducted under conditions of constant volume (NVT ensemble) and pressure (NPT ensemble), respectively.

_{av}= 100 system at φ = 0.40 under full confinement and 100-chain N

_{av}= 12 system at φ = 0.48 with confinement imposed on the short dimension and periodic boundary conditions enforced on the long ones. In both cases, the wall thickness is approximately equal to the monomer diameter, effectively corresponding to 2-D thin films. Successive steps include compaction of the athermal HS chain systems to even higher volume fractions or activation of SW attractive interactions under constant volume (chains) or constant pressure (monomers).

**r**

_{i}, the (minimum) CCE norms with respect to all tested reference crystals, ${\epsilon}_{i}^{X}$, and optionally the orientation(s) of the corresponding symmetry axi(e)s. Additionally, a pair of pdb/psf files is provided to be used for successive visualization through appropriate software (for example VMD [5], as used in the present work).

## 3. Results

#### 3.1. CCE Norm Application to Perfect Crystals

^{FCC}= 2.44 × 10

^{−4}and ε

^{BCC}= 1.90 × 10

^{−4}) but non-zero. This is because both crystals belong to the cubic system and the fingerprint corresponds to a set of four threefold roto-inversion axes. The determination of the orientation of the first axis follows the same procedure as for the other crystals but another search (plus discretization) in space is required to establish the orientation of the second axis, as explained in the methods section. Accordingly, for a given mesh size (0.1 rad) it is more difficult to find the optimal set of axes for the BCC and FCC cases than the other crystals. Unavoidably, these crystals will have higher uncertainty in their detection compared to the other ordered motifs. However, we should note that the established values are already 10

^{3}times lower than the identification threshold, ε

^{FCC}= 0.245.

#### 3.2. CCE Norm Application to Computer-Generated, 3-D Bulk Systems

_{1}in Equation (2)). The aim of the present work is not to present new physics on 2-D (thin film) or 3-D (bulk) crystallization of athermal or attractive chains but rather to demonstrate the efficiency of the CCE algorithm to accurately identify local structure and to detect and quantify crystal nucleation and growth in general particulate and atomic systems. Initial configurations correspond to bulk, 3-D HS chains at low packing density (φ = 0.05), as can be seen in Figure A1 of the Appendix A.

^{X}< ε

^{thres}= 0.245. Throughout all images and snapshots to be presented in the continuation the coloring scheme is the same: red, blue, green, purple and cyan colors correspond to HCP, FCC, FIV, HEX and BCC crystals, respectively. All sites that show no similarity for none of the reference crystals are labeled as “amorphous”, or more precisely as “none of the above”, and are represented in yellow. In parallel, to render the depth of the snapshot visually accessible, crystal-like and amorphous sites are shown with reduced dimensions in a 3:5 and 1:5 scale, respectively. In the specific examples, the established ordered morphologies, as identified by the CCE algorithm, are almost perfect HCP and FCC crystals for the SW chain systems at dilute conditions (Figure 6) while HEX and BCC are the dominant ones for the SW monomeric clusters in Figure 7. A detailed analysis on the phase behavior of chains and monomers in the whole spectrum of the intensity and range parameters as well as an explanation on why certain ordered morphologies are favored for specific simulation conditions will be presented in a future work. A significant difference between the low-density chain crystals, as obtained from the NVT simulations, and the high-density ones for monomers, as obtained from the NPT simulations, is the presence in the former of an external surface of unidentified character that surrounds the core of the cluster as seen in the snapshots of Figure 6. Atoms on the surface of the cluster cannot be assigned to any reference crystal because they lack a complete set of first neighbors, and hence have incomplete Voronoi polyhedra [41].

^{FCC}= 0.0223), followed by HCP-like (ε

^{HCP}= 0.0340), HEX-like (ε

^{HEX}= 0.0466) and BCC-like (ε

^{BCC}= 0.0536), the latter being approximately 2.4 times higher than the minimum recorded for FCC. All these CCE norm values are very small, close to zero, and significantly lower than the threshold value of 0.245.

_{coord}(X) ones are taken into account for the calculation of the X-CCE norm.

^{HCP}= 0.050 (left-most panel), 0.10, 0.20 and 0.30 (right-most panel). The lowest HCP-CCE norm is linked to an almost perfect HCP ordered structure. As the norm increases small structural defects appear, including minor radial and/or orientational deviations. This is most evident in the rightmost snapshot with ε

^{HCP}= 0.30. This value is actually higher than the threshold (0.245) used to assign HCP character. Accordingly, based on the HCP-CCE norms in Figure 9, all structures are identified as HCP-like except the one on the right-most panel. Increasing (weakening) the criterion for the detection to a value higher than 0.30 would also include this structure in the HCP population. However, caution should be exercised as too large an increase in ε

^{thres}may result in sites having dual or multiple similarity eliminating the ability of the CCE method to discriminate between competing, structurally similar crystals.

_{coord}(X), volume, V

_{VP}(X), surface area, A

_{VP}(X), number of faces, F(X), number of edges, E(X) and number of vertices, V(X). Furthermore, local packing density can be calculated as:

_{1}/2, is connected to packing density through:

_{VP}(X), A

_{VP}(X) and Q(X) between the computer-generated structures of high crystal similarity and the reference ones, are summarized in Table 7. A remarkable trend, which must be studied in more detail through an extended sample of configurations, is that the generated HEX and BCC crystals are significantly more spacious than the perfect analogs. In parallel, the standard isoperimetric quotient seems to be in very good (3% for HEX) to an almost perfect (0.01% for FCC) agreement between the generated structures and the analogous perfect templates. The computer-generated BCC structure is still, with a slightly reduced margin compared to the perfect templates, the local structure with the highest isoperimetric quotient value, even larger than the ones of the close packed HCP and FCC crystals (Q(BCC) > Q(HCP) ≈ Q(FCC) > Q(HEX)).

_{SW}= 2.1 and σ

_{2}= 1.2). Dashed horizontal and vertical lines denote the threshold (ε

^{thres}= 0.245) below which a site is identified as Y-like and X-like, respectively. The discrimination ability of the CCE algorithm is demonstrated by the empty square whose borders are defined by the points (0, 0), (0, ε

^{thres}), (ε

^{thres}, 0) and (ε

^{thres}, ε

^{thres}). In practice, absence of data points in this area implies that no site possesses dual ordered character by having simultaneously low CCE norm with respect to the two different X and Y reference crystals.

#### 3.3. CCE Application to Computer-Generated, 2-D Thin-Film Systems

_{wall}→ σ

_{1}). Local packing density, ${\rho}_{n}^{*}\left(X\right)$, is defined here as the inverse of the surface area, A

_{VP}(X), of the Voronoi polygon of crystal X:

_{wall}→ σ

_{1}) we get:

_{VP}(X) and P

_{VP}(X) are the surface area and the perimeter of the Voronoi polygon for the 2-D crystal X the standard isoperimetric ratio, q(X), is defined as [54]:

_{av}= 100 system under full confinement (left panel in Figure 11) is characterized by low density (φ = 0.40) and equivalently low surface coverage (${\phi}^{*}$ = 0.60). As a consequence, only a very small fraction of sites shows ordered structure. As density increases (right panel in Figure 11 corresponding to 100-chains of N

_{av}= 12 at φ = 0.48 or equivalently ${\phi}^{*}$ = 0.472) the population of sites with crystal local structure increases appreciably, especially that of the TRI character.

_{2}= 1.3) shows a clear tendency for close packing, as most sites adopt a triangular structure, which possesses the highest density among all 2-D crystals. The snapshot of the right panel (ε = 0.5 and σ

_{2}= 1.6) also presents higher degree of ordering than the original athermal packing (right panel of Figure 11) and different mixture of crystal sites. Here, the population of square-like sites is comparable to that of triangular ones. In both cases, no honeycomb structures are detected, an expected trend given that the honeycomb crystal shows significantly lower surface coverage than the square and triangular ones. Additionally, in the configuration of mixed character (TRI and SQU) there is a small, but non-zero population of sites with non-crystal, pentagonal local symmetry. In contrast, in the TRI-dominated system, pentagonal sites are completely absent. A systematic analysis of the phase behavior as a function of attraction intensity and range in thin films of SW chains as well as an interpretation of the observed tendencies will be presented in a future work.

_{av}= 12 system (which can be seen in Figure 11, right panel) with progressively reduced similarity to triangular crystal as quantified by the TRI-CCE norm. From left to right the values are approximately 0.05, 0.10, 0.20 and 0.30. The last structure with ε

^{TRI}= 0.30 is not recognized as TRI-like, as the norm is higher than the threshold value of ε

^{thres}= 0.245. Visual inspection qualitatively confirms the diminishing similarity as established quantitatively by the CCE descriptor. This is further demonstrated by the data on the statistics of the Voronoi polygon, as reported in Table 9, where they are further compared against the corresponding one of the perfect, reference triangular crystal. It can be deduced that the computer-generated local structure with the closest TRI similarity differs by approximately 5.8, 2.9 and 0.2% in surface area, perimeter and standard isoperimetric ratio with respect to the reference crystal. For the least similar structure the corresponding percentages (of the ones reported in Table 9) increase to 15.5, 9.0 and 2.9%, for A

_{VP}(TRI), P

_{VP}(TRI) and q(TRI), respectively. In both cases and as expected, the computer-generated polygons are larger than the reference ones and thus less dense.

^{thres}= 0.245), means that no site exists with dual crystal character. Out of all possible comparisons of crystal pairs (= N

_{at}× N

_{s,2}! = 1200 × 6 = 7200) there is only one site which is detected with dual character (ε

^{TRI}, ε

^{HON}< ε

^{thres}) as seen in the left panel of Figure 14, with the HON similarity (ε

^{HON}≈ 0.24) being the weakest one and very close to the detection threshold. In the very rare case of a site having two CCE norms lower than the threshold, it adopts the one that has the lower value.

## 4. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

BCC | Body Centered Cubic |

CCE | Characteristic Crystallographic Element (norm) |

CAN | Common Neighbor Analysis |

FCC | Face Centered Cubic |

FIV | Fivefold |

HCP | Hexagonal Close Packed |

HEX | Hexagonal |

HON | Honeycomb |

HS | Hard Sphere |

MC | Monte Carlo |

RCP | Random Close Packing |

PEN | Pentagonal |

RHCP | Random Hexagonal Close Packed |

SQU | Square |

SSP | Short-Range Order Symmetry Parameter |

SW | Square Well |

TRI | Triangular |

## Appendix A

**Figure A1.**Snapshots of equilibrated packings of linear hard-sphere chains in the bulk at a packing density of 0.05, used as initial configurations for successive simulations with the square well potential. (

**a**) 100 chains of N

_{av}= 12; (

**b**) 50 chains of N

_{av}= 24. Spheres are colored according to the identity of the parent chain. Coordinates of sphere centers are subjected to periodic boundary conditions in all dimensions. Each figure panel is also available as stand-alone image in 3-D, interactive format in the Supplementary Materials.

_{av}= 100 system at φ = 0.05 is shown under confinement in all dimensions. The MC protocol to produce such polymer systems under extreme confinement is described in [49,57,58]. The right panel hosts a snapshot of a 100-chain N

_{av}= 12 system at φ = 0.48. Here, periodic boundary conditions are applied on the long dimensions of the cell while flat parallel, impenetrable walls exist in the short dimension. In both systems of Figure A2 the thickness in the short (x) dimension corresponds to one layer, i.e., (d

_{wall}→ σ

_{1}).

**Figure A2.**Snapshots of equilibrated packings of linear, hard-sphere chains used as initial configurations for successive simulations with the hard sphere or square well potential. (

**a**) 48 chains of N

_{av}= 100 at φ = 0.05 under confinement in all dimensions; (

**b**) 100 chains of N

_{av}= 12 at φ = 0.48 with periodic boundary conditions in the long dimensions and confinement in the short one. Spheres are colored according to the parent chain. Each figure panel is also available as stand-alone image in 3-D, interactive format in the Supplementary Materials.

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**Figure 1.**Reference 3-D crystals used for the demonstration of the Characteristic Crystallographic Element (CCE) descriptor. From left to right: hexagonal close packed (HCP), face centered cubic (FCC), hexagonal (HEX) and body centered cubic (BCC) lattices. Color convention according to which HCP, FCC, non-crystallographic fivefold (FIV) (not shown here), HEX and BCC sites are represented in blue, red, green, purple and cyan is used throughout the present manuscript. Image created with the Visual Molecular Dynamics (VMD) software [5]. Each figure panel is also available as stand-alone image in 3-D, interactive format in the Supplementary Material.

**Figure 2.**Reference 2-D crystals used for the demonstration of the CCE descriptor. From left to right: triangular (TRI), square (SQU) and honeycomb (HON). The color convention, according to which TRI, SQU, PEN (not shown here) and HON sites are represented in blue, red, green and cyan is used throughout the present manuscript. Image created with the VMD software. Each figure panel is also available as stand-alone image in 3-D, interactive format in the Supplementary Material.

**Figure 3.**Voronoi polyhedron and nearest neighbors for a given site (red) in the 3-D reference crystals. From left to right: HCP, FCC, HEX and BCC. Voronoi edges are shown as green lines, and tangent neighbor sites in blue. Sphere radii have been reduced by 2:5 for visualization purposes. Image created with the VMD software. Each figure panel is also available as stand-alone image in 3-D, interactive format in the Supplementary Material.

**Figure 4.**Voronoi polygon and nearest neighbors for 2-D reference crystals. From left to right: TRI, SQU and HON. Voronoi edges are shown as green lines, neighbor sites in blue and reference sphere in red. Image created with the VMD software. Each figure panel is also available as stand-alone image in 3-D, interactive format in the Supplementary Material.

**Figure 5.**Value of the FCC-CCE norm (ε

^{FCC}), when applied on a local environment corresponding to a perfect FCC crystal, as a function of the step of the mesh discretization used to explore the orientation of the first and second three-fold roto-inversion symmetry axes.

**Figure 6.**Snapshots at the end of NVT MC simulations on SW chains (N

_{av}= 24). (

**a**) ε = 0.5 and σ

_{2}= 2.0; (

**b**) ε = 0.5 and σ

_{2}= 2.3. Spheres are color-coded according to the X-crystal similarity (ε

^{X}< ε

^{thres}= 0.245) as identified by the value of the CCE norm. Blue, red, green, purple and cyan colors correspond to HCP, FCC, FIV, HEX and BCC crystals, respectively. Crystal sites are shown with reduced diameter in a 3:5 scale. Amorphous (or unidentified) sites are shown in yellow color and reduced (1:5) diameter for visualization purposes. Image created with the VMD software. Each figure panel is also available as stand-alone image in 3-D, interactive format in the Supplementary Materials.

**Figure 7.**Snapshots at the end of NPT MC simulations on SW monomers. (

**a**) ε = 2.1 and σ

_{2}= 1.6; (

**b**) ε = 2.7 and σ

_{2}= 1.3. Spheres are color-coded according to the X-crystal similarity (ε

^{X}< ε

^{thres}= 0.245) as identified by the value of the CCE norm. Blue, red, green, purple and cyan colors correspond to HCP, FCC, FIV, HEX and BCC crystals, respectively. Crystal sites are shown with reduced diameter in a 3:5 scale. Amorphous (or ”none of the above”) sites are shown in yellow color and reduced (1:5) diameter for visualization purposes. Image created with the VMD software. Each figure panel is also available as stand-alone image in 3-D, interactive format in the Supplementary Materials.

**Figure 8.**Snapshots of local environments, as generated through present MC simulations, showing very high similarity to a specific reference X-crystal, as quantified by the corresponding very low value of the X-CCE norm. From left to right: HCP-like (ε

^{HCP}= 0.0340); FCC-like (ε

^{FCC}= 0.0223); HEX-like (ε

^{HEX}= 0.0466); BCC-like (ε

^{BCC}= 0.0536). Reference site and nearest neighbors are shown in red and blue, respectively. Also shown are the corresponding Voronoi polyhedra as green lines. Sites are shown with reduced diameter in a 2:5 scale for visualization purposes. Image created with the VMD software. Each figure panel is also available as stand-alone image in 3-D, interactive format in the Supplementary Materials.

**Figure 9.**Computer-generated, local structures with progressively reduced HCP-similarity and thus with increasingly high HCP-CCE norm as quantified by the CCE descriptor. From left to right: ε

^{HCP}= 0.05, 0.10, 0.20 and 0.30. The latter case is not characterized as HCP-like since ε

^{HCP}> ε

^{thres}= 0.245. Reference site and nearest neighbors are shown in red and blue, respectively. The corresponding Voronoi polyhedra are also shown as green lines. Sites are shown with reduced diameter in a 2:5 scale for visualization purposes. Image created with the VMD software. Each figure panel is also available as stand-alone image in 3-D, interactive format in the Supplementary Materials.

**Figure 10.**All possible combinations of parity plots X- vs. Y-CCE norm (X ≠ Y) for the final configuration of NPT MC simulations on a system of N

_{at}= 1200 monomers interacting with the square well potential (ε

_{SW}= 2.1 and σ

_{2}= 1.2). Top-left: X = HCP and Y = FCC/FIV/HEX/BCC; Top-right: X = FCC and Y = FIV/HEX/BCC; Bottom-left: X = FIV and Y = HEX/BCC; Bottom-right: X = HEX and Y = BCC. Vertical and horizontal dashed lines denote the CCE norm threshold below which a site is labeled as X- and Y-like, respectively. Blue, red, green, purple, cyan and yellow colors correspond to HCP-, FCC-, FIV-, HEX-, BCC-like and amorphous (or unidentified) sites, respectively.

**Figure 11.**Snapshots of MC simulations on thin-films of linear, hard-sphere chains. (

**a**) 48-chain N

_{av}= 100 at φ = 0.40 under confinement in all dimensions. (

**b**) 100-chain N

_{av}= 12 at φ = 0.40 with confinement in the short dimension and periodic boundary conditions in the long ones. In both cases inter-wall distance in the small dimension corresponds to a single layer. Spheres are color-coded according to the X-crystal similarity (ε

^{X}< ε

^{thres}= 0.245) as identified by the value of the CCE norm. Blue, red, green and cyan colors correspond to TRI, SQU, PEN and HON symmetries, respectively. Amorphous (or unidentified) sites are shown in yellow color. Image created with the VMD software. Each figure panel is also available as stand-alone image in 3-D, interactive format in the Supplementary Materials.

**Figure 12.**Snapshots of MC simulations on thin films of 100 chains (N

_{av}= 12). Spheres interact with the SW potential. Confinement is applied on the short dimension and periodic boundary conditions on the long ones. (

**a**) ε = 0.5 and σ

_{2}= 1.3. (

**b**) ε = 0.5 and σ

_{2}= 1.6. Spheres are color-coded according to the X-crystal similarity (ε

^{X}< ε

^{thres}= 0.245) as identified by the value of the CCE norm. Blue, red, green and cyan colors correspond to TRI, SQU, PEN and HON symmetries, respectively. Amorphous (or “none of the above”) sites are shown in yellow color. Image created with the VMD software. Each figure panel is also available as stand-alone image in 3-D, interactive format in the Supplementary Materials.

**Figure 13.**Computer-generated, local structures with progressively reduced TRI-similarity and thus with increasingly high TRI-CCE norm. From left to right: ε

^{TRI}= 0.05, 0.10, 0.20 and 0.30. The latter case is not characterized as TRI-like since ε

^{TRI}> ε

^{thres}= 0.245. Reference site and nearest neighbors are shown in red and blue, respectively. Image created with the VMD software. Each figure panel is also available as stand-alone image in 3-D, interactive format in the Supplementary Materials.

**Figure 14.**All possible combinations of parity plots X- vs. Y-CCE norm (X ≠ Y) from NVT simulations on a (2-D) thin-film system of 100-chains of N

_{av}= 12 at φ = 0.48 as seen in the right panel of Figure 11. (

**a**) X = TRI and Y = SQU/PEN/HON; (

**b**) X = SQU and Y = PEN/HON; (

**c**) X = PEN and Y = HON; Vertical and horizontal dashed lines denote the CCE norm threshold below which a site is labeled as X- and Y-like, respectively. Blue, red, green, cyan and yellow colors correspond to TRI-, SQU-, PEN-, HON-, and amorphous (or unidentified) sites, respectively.

**Table 1.**Number of nearest neighbors, N

_{coord}(X), number of distinct geometric symmetry elements N

_{el}(X), along with their type, number of actions per element, N

_{act}(k,X) and their type for the 3-D crystals (X = HCP, FCC, HEX and BCC) as well as the fivefold (FIV) local symmetry.

Reference Structure → Fingerprint ↓ | HCP | FCC | HEX | BCC | FIV |
---|---|---|---|---|---|

N_{coord}(X) | 12 | 12 | 8 | 8 | 12 |

N_{el}(X) | 1 | 5 | 2 | 5 | 2 |

Geometric Symmetry Element k$\mathbf{(}\mathit{k}\mathbf{\in}\left[\mathbf{1}\mathbf{,}{\mathit{N}}_{\mathit{e}\mathbf{l}}\left(\mathit{X}\right)\right]\mathbf{)}$ | (k = 1) Roto-inversion Axis | (k = 1,…,4) Roto-inversion Axes (k = 5) Inversion Center | (k = 1) Rotation Axis (k = 2) Inversion Center | (k = 1,…,4) Roto-inversion Axes (k = 5) Inversion Center | (k = 1) Rotation Axis (k = 2) Inversion Center |

N_{act}(k,X) | (k = 1) 5 | (k = 1,…,4) 5 (k = 5) 1 | (k = 1) 5 (k = 2) 1 | (k =1,…,4) 5 (k = 5) 1 | (k = 1) 4 (k = 2) 1 |

Symmetry Actions of Geometric Element k | (k = 1) ${\overline{6}}_{c}^{1}$, ${\overline{6}}_{c}^{2}$, ${\overline{6}}_{c}^{3}$, ${\overline{6}}_{c}^{4}$, ${\overline{6}}_{c}^{5}$ | (k = 1,…,4) ${\overline{3}}_{c}^{1}$, ${3}_{c}^{2}$, ${\overline{3}}_{c}^{3}$, ${\overline{3}}_{c}^{4}$, ${\overline{3}}_{c}^{5}$ (k = 5) $\overline{1}$ | (k = 1) ${6}_{c}^{1}$, ${6}_{c}^{2}$, ${6}_{c}^{3}$, ${6}_{c}^{4}$, ${6}_{c}^{5}$ (k = 2) $\overline{1}$ | (k = 1,…,4) ${\overline{3}}_{c}^{1}$, ${3}_{c}^{2}$, ${\overline{3}}_{c}^{3}$, ${\overline{3}}_{c}^{4}$, ${\overline{3}}_{c}^{5}$ (k = 5) $\overline{1}$ | (k = 1) ${5}_{c}^{1}$, ${5}_{c}^{2}$, ${5}_{c}^{3}$, ${5}_{c}^{4}$, ${5}_{c}^{5}$ (k = 2) $\overline{1}$ |

**Table 2.**Number of nearest neighbors, N

_{coord}(X), number of distinct geometric symmetry elements N

_{el}(X), along with their type, number of actions per element, N

_{act}(k,X) and their type for the 2-D crystals (X = TRI, SQU and HON) as well as the pentagonal (PEN) local symmetry.

Reference Structure → Fingerprint ↓ | TRI | SQU | HON | PEN |
---|---|---|---|---|

N_{coord}(X) | 6 | 4 | 3 | 5 |

N_{el}(X) | 2 | 2 | 1 | 1 |

Geometric Symmetry Element k$\mathbf{(}\mathit{k}\mathbf{\in}\left[\mathbf{1}\mathbf{,}{\mathit{N}}_{\mathit{e}\mathbf{l}}\left(\mathit{X}\right)\right]\mathbf{)}$ | (k = 1) Rotation Axis (k = 2) Inversion Center | (k = 1) Rotation Axis (k = 2) Inversion Center | (k = 1) Rotation Axis | (k = 1) Rotation Axis |

N_{act}(k,X) | (k = 1) 5 (k = 2) 1 | (k = 1) 3 (k = 2) 1 | (k = 1) 2 | (k = 1) 4 |

Symmetry Actions of Geometric Element k | (k = 1) ${6}_{c}^{1}$, ${6}_{c}^{2}$, ${6}_{c}^{3}$, ${6}_{c}^{4}$, ${6}_{c}^{5}$ (k = 2) $\overline{1}$ | (k = 1) ${4}_{c}^{1}$, ${4}_{c}^{2}$, ${4}_{c}^{3}$ (k = 2) $\overline{1}$ | (k = 1) ${3}_{c}^{1}$, ${3}_{c}^{2}$, | (k = 1) ${5}_{c}^{1}$, ${5}_{c}^{2}$, ${5}_{c}^{3}$, ${5}_{c}^{4}$ |

**Table 3.**Characteristic Crystallographic Element (CCE) norm, ε

^{X}, when the analysis is applied on reference 3-D hexagonal close packed (HCP), face centered cubic (FCC), hexagonal (HEX) and body centered cubic (BCC) crystals. Also reported are results from CCE application with respect to the fivefold (FIV) local symmetry.

ε^{X} →Reference Lattice ↓ | HCP | FCC | HEX | BCC | FIV |
---|---|---|---|---|---|

HCP | 0 | 0.257 | 0.290 | 0.412 | 0.228 |

FCC | 0.246 | 0.000244 | 0.377 | 0.518 | 0.229 |

HEX | 0.280 | 0.239 | 0 | 0.239 | 0.185 |

BCC | 0.284 | 0.165 | 0.246 | 0.000190 | 0.276 |

**Table 4.**Characteristic Crystallographic Element (CCE) norm, ε

^{X}, when the analysis is applied on reference 2-D honeycomb (HON), square (SQU) and trigonal (TRI) lattices. Also reported are results from CCE application with respect to the pentagonal local symmetry (X = PEN).

ε^{X} →Reference Lattice ↓ | TRI | SQU | HON | PEN |
---|---|---|---|---|

TRI | 0 | 0.275 | 0.699 | 0.397 |

SQU | 0.301 | 0 | 0.613 | 0.318 |

HON | 0.649 | 0.465 | 0 | 0.420 |

**Table 5.**Statistics of the Voronoi polyhedron of perfect, reference 3-D crystal X: coordination number N

_{coord}(X), number of faces F(X), vertices V(X) and edges E(X), volume V

_{VP}(X), surface area A

_{VP}(X), local number density ρ

_{n}(X), packing density φ(X) and standard isoperimetric quotient Q(X).

X | N_{coord}(X) | F(X) | V(X) | E(X) | V_{VP}(X) | A_{VP}(X) | ρ_{n}(X) | φ(X) | Q(X) |
---|---|---|---|---|---|---|---|---|---|

HCP | 12 | 12 | 14 | 14 | 0.7071 | 4.243 | 1.414 | 0.7404 | 0.7405 |

FCC | 12 | 12 | 14 | 24 | 0.7071 | 4.243 | 1.414 | 0.7404 | 0.7405 |

HEX | 8 | 8 | 12 | 18 | 0.8660 | 5.196 | 1.155 | 0.6046 | 0.6045 |

BCC | 8 | 14 | 24 | 36 | 0.7698 | 4.464 | 1.299 | 0.6800 | 0.7534 |

X-like | N_{coord}(X) | F(X) | V(X) | E(X) | V_{VP}(X) | A_{VP}(X) | ρ_{n}(X) | φ(X) | Q(X) |
---|---|---|---|---|---|---|---|---|---|

HCP | 12 | 13 | 22 | 33 | 0.7534 | 4.426 | 1.327 | 0.6950 | 0.7404 |

FCC | 12 | 13 | 22 | 33 | 0.7311 | 4.339 | 1.368 | 0.7162 | 0.7400 |

HEX | 8 | 15 | 26 | 39 | 1.038 | 5.806 | 0.9634 | 0.5044 | 0.6226 |

BCC | 8 | 14 | 24 | 36 | 0.9214 | 5.045 | 1.085 | 0.5683 | 0.7478 |

**Table 7.**Relative percentage difference between the computer-generated local structures with very high crystal similarity (as visualized in Figure 8) and the reference, perfect 3-D crystals with respect to volume, surface area and standard isoperimetric quotient of the enclosing Voronoi polyhedron. Also reported in the last column is the corresponding X-CCE norm for the computer-generated structure.

X | % V_{VP}(X) | % A_{VP}(X) | % Q(X) | X-CCE norm |
---|---|---|---|---|

HCP | 6.55 | 4.31 | 0.0135 | 0.0340 |

FCC | 3.39 | 2.26 | 0.0675 | 0.0223 |

HEX | 19.9 | 10.5 | 2.99 | 0.0466 |

BCC | 19.7 | 13.0 | 0.743 | 0.0536 |

**Table 8.**Statistics of the Voronoi polygon of perfect, reference 2-D crystal X: coordination number N

_{coord}(X), number of vertices V(X) and edges E(X), surface area A

_{VP}(X), perimeter P

_{VP}(X), local packing density ${\rho}_{n}^{*}\left(X\right)$, surface coverage ${\phi}^{*}\left(X\right)$ and standard isoperimetric ratio q(X).

X | N_{coord}(X) | V(X) | E(X) | A_{VP}(X) | P_{VP}(X) | ${\mathit{\rho}}_{\mathit{n}}^{*}\left(\mathit{X}\right)$ | ${\mathit{\phi}}^{*}\left(\mathit{X}\right)$ | q(X) |
---|---|---|---|---|---|---|---|---|

TRI | 6 | 6 | 6 | 0.8660 | 3.464 | 1.155 | 0.907 | 0.907 |

SQU | 4 | 4 | 4 | 1.000 | 4.000 | 1.000 | 0.785 | 0.785 |

HON | 3 | 3 | 3 | 1.299 | 5.196 | 0.7698 | 0.604 | 0.604 |

**Table 9.**Statistics of the Voronoi polygon of computer-generated local structures, visualized in Figure 13, with progressively poor triangular singularity as quantified by the corresponding CCE norm, ε

^{TRI}: coordination number N

_{coord}(X), number of vertices V(X) and edges E(X), surface area A

_{VP}(X), perimeter P

_{VP}(X), local packing density ${\rho}_{n}^{*}\left(X\right)$, surface coverage ${\phi}^{*}\left(X\right)$ and standard isoperimetric ratio q(X). Also shown for comparison in the first row are the results that correspond to the perfect, reference 2-D triangular crystal.

ε^{TRI} | N_{coord}(TRI) | V(TRI) | E(TRI) | A_{VP}(TRI) | P_{VP}(TRI) | ${\mathit{\rho}}_{\mathit{n}}^{*}\left(\mathbf{TRI}\right)$ | ${\mathit{\phi}}^{*}\left(\mathbf{TRI}\right)$ | q(TRI) |
---|---|---|---|---|---|---|---|---|

0.00 | 6 | 6 | 6 | 0.8660 | 3.464 | 1.155 | 0.907 | 0.907 |

0.05 | 6 | 6 | 6 | 0.9160 | 3.564 | 1.093 | 0.858 | 0.905 |

0.10 | 6 | 6 | 6 | 0.9420 | 3.620 | 1.062 | 0.834 | 0.903 |

0.20 | 6 | 6 | 6 | 1.006 | 3.766 | 0.9940 | 0.781 | 0.891 |

0.30 | 6 | 6 | 6 | 1.000 | 3.776 | 1.000 | 0.786 | 0.881 |

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**MDPI and ACS Style**

Ramos, P.M.; Herranz, M.; Foteinopoulou, K.; Karayiannis, N.C.; Laso, M.
Identification of Local Structure in 2-D and 3-D Atomic Systems through Crystallographic Analysis. *Crystals* **2020**, *10*, 1008.
https://doi.org/10.3390/cryst10111008

**AMA Style**

Ramos PM, Herranz M, Foteinopoulou K, Karayiannis NC, Laso M.
Identification of Local Structure in 2-D and 3-D Atomic Systems through Crystallographic Analysis. *Crystals*. 2020; 10(11):1008.
https://doi.org/10.3390/cryst10111008

**Chicago/Turabian Style**

Ramos, Pablo Miguel, Miguel Herranz, Katerina Foteinopoulou, Nikos Ch. Karayiannis, and Manuel Laso.
2020. "Identification of Local Structure in 2-D and 3-D Atomic Systems through Crystallographic Analysis" *Crystals* 10, no. 11: 1008.
https://doi.org/10.3390/cryst10111008