Common neighbor analysis¶

CommonNeighborAnalysis gives each atom the crystal structure of its neighborhood: fcc, hcp, bcc, icosahedral or other.

import numpy as np
import mdapy as mp
from mdapy.build import build_crystal
from mdapy.identify import CommonNeighborAnalysis

# fcc copper of 4 x 4 x 4 cubic cells
a = 3.615
system = build_crystal("fcc", a, elements="Cu", repeat=(4, 4, 4))

result = CommonNeighborAnalysis().compute(system)  # adaptive
result.counts()  # {'other': 0, 'fcc': 256, 'hcp': 0, 'bcc': 0, 'ico': 0}
system = system.with_columns(result)  # the column 'structure'

fixed = CommonNeighborAnalysis(cutoff=0.854 * a).compute(system)  # conventional

In a polycrystal the atoms that are not fcc mark the grain boundaries:

from mdapy.build import build_crystal, build_polycrystal
from mdapy.visualization import render, write_png

copper = build_crystal("fcc", 3.615, elements="Cu")
grains = build_polycrystal(copper, mp.Box(60.0), 6, seed=4)
structures = CommonNeighborAnalysis().compute(grains)
whole = render(grains, color=structures, width=600, height=600, legend=True)
fcc = structures.structure == 1
boundaries = render(
    grains, atoms=~fcc, color=structures, width=600, height=600, legend=True
)
write_png("images/cna.png", np.hstack([whole, boundaries]))

Common neighbor analysis of a polycrystal

A copper polycrystal of six grains as built, colored by structure: fcc green, other white. Right: the same without the fcc atoms, which leaves the network of grain boundaries.

Parameters¶

Name	Type	Default	Unit	Meaning
`cutoff`	float64	`None`	Å	cutoff of the neighbors and of the bonds between them; without it, each atom has its own cutoff (adaptive analysis)

compute(system, *, neighbors=None, progress=None, cancel=None) takes the arguments of every analysis. A given neighbor list must reach the cutoff or, without a cutoff, hold the 14 nearest neighbors of every atom.

Results¶

A StructureResult with the types

Index	Name	Default color
0	`other`	light grey
1	`fcc`	green
2	`hcp`	red
3	`bcc`	blue
4	`ico`	yellow

Errors¶

The constructor raises InvalidParameter for a cutoff that is not positive; compute raises it for a neighbor list that does not fit.

Definition¶

Each bond from an atom to one of its neighbors gets a signature of three numbers: the number of neighbors the two atoms have in common, the number of bonds among those common neighbors, and the number of bonds in the largest connected group those bonds form. An atom with 12 neighbors is fcc when all 12 signatures are (4, 2, 1), hcp when six are (4, 2, 1) and six (4, 2, 2), and icosahedral when all are (5, 5, 5). An atom with 14 neighbors is bcc when eight signatures are (6, 6, 6) and six (4, 4, 4). Any other atom is other.

Without cutoff, the analysis is adaptive: each atom has its own cutoff from its nearest neighbors, so strained or hot crystals of any lattice constant need no parameter. For fcc, hcp and icosahedral order it takes the 12 nearest neighbors \(r_1 \le \dots \le r_{12}\) and the cutoff

\[ r_c = \frac{1 + \sqrt 2}{2}\, \frac{1}{12} \sum_{j=1}^{12} r_j , \]

halfway between the first and second shells of fcc; neighbors and bonds between them count when they are no longer than \(r_c\). When that does not give fcc, hcp or icosahedral, it takes the 14 nearest neighbors for bcc and

\[ r_c = \frac{1 + \sqrt 2}{2}\, \frac{1}{14} \Big(\frac{2}{\sqrt 3} \sum_{j=1}^{8} r_j + \sum_{j=9}^{14} r_j\Big). \]

With cutoff, the conventional analysis, the neighbors of an atom are those closer than the cutoff, and an atom must have exactly 12 or 14 of them; the same cutoff decides the bonds between neighbors. A cutoff between the first and second shells is \(0.854\,a\) for fcc and \(1.207\,a\) for bcc of lattice constant \(a\).

In a system without periodic directions and fewer than 15 atoms, the neighbors of each atom are all the others.

Common neighbor analysis¶

Parameters¶

Results¶

Errors¶

Definition¶

See also¶